HOMOTOPY EQUIVALENCE AND DONALDSON INVARIANTS WHEN b^{+} = 1. I: COBORDISMS OF MODULI SPACES AND CONTINUITY OF

GLUING MAPS ^{1}

PAUL M. N. FEEHAN AND THOMAS G. LENESS

Abstract. The present article is the first in a series whose ultimate goal is to prove the Kotschick-Morgan conjecture concerning the wall-crossing formula for the Donaldson invariants of a four-manifold with b^{+} = 1. The conjecture asserts that, in essence, the wallcrossing terms due to changes in the metric depend at most on the homotopy type of the four-manifold and the degree of the invariant. Our principal interest in this conjecture is due to (i) the fact that its proof is expected to resemble that of an important intermediate step towards a proof of Witten’s conjecture concerning the relation between Donaldson and Seiberg-Witten invariants [63], using PU(2) monopoles as described in [20], [17], and (ii) it affords us another venue in which to address some of the technical difficulties arising in our work on Witten’s conjecture. The additional difficulties in the case of PU(2) monopoles are due to the considerably more complicated gluing theory, the presence of obstructions to deformation and gluing, and the need to consider links of positive-dimensional families of ‘reducibles’ even in the presence of ‘simple type’ assumptions. Consequently, it is extremely useful to examine the simpler case of the Kotschick-Morgan conjecture in parallel and separately take into account the difficulties that arise in the case of Witten’s conjecture. Witten’s conjecture should follow from a final step analogous to Göttsche’s computation of the wall-crossing terms, assuming that the Kotschick-Morgan conjecture holds [27].

In this article we present the proof of the first half of a general gluing theorem endowed with several important enhancements not present in previous treatments due to Taubes, Donaldson, and Mrowka. We prove that the Uhlenbeck-compactified moduli spaces of instantons on S^{4} for two nearby metrics, which a priori need only be cobordant, are actually homeomorphic and diffeomorphic on smooth strata. We also show that the gluing maps, with obstructions to deformation permitted, extend to maps on the closure of the gluing data which are continuous with respect to the Uhlenbeck topology. The second half of the proof of our general gluing theorem is proved in the sequel [14].

We note that Kronheimer and Mrowka have conjectured a relation between the Yang-Mills and Seiberg-Witten Floer homologies for certain three-manifolds Y [37], [39], [48] and they have a recently announced a program to prove this conjecture using moduli spaces of PU(2) monopoles over cylinders Y × ℝ. One of the difficulties in their program, which concerns the contributions of Seiberg-Witten solutions in the presence of instanton bubbling, to some extent mirrors those in our four-manifold program discussed in [20], [21], [17], [18], [19]. Their work is expected to ultimately lead to a proof that all knots in S^{3} have ‘property P ’.

1Submitted to a print journal, December 10, 1998. math.DG/yymmnnn. The first author was supported in part by an NSF Mathematical Sciences Postdoctoral Fellowship under

grant DMS 9306061 and by NSF grants DMS 9704174 and, through the Institute for Advanced Study, by DMS 9729992.

Contents

1. Introduction 2 2. The wall-crossing formula for Donaldson invariants 10 3. Splicing connections 21 4. Regularity of solutions to quasi-linear systems 30 5. Decay estimates for curvature and connections in Coulomb and radial gauges 39 6. Eigenvalue estimates for Laplacians on self-dual two-forms 49 7. Existence of gluing maps and obstruction sections 61 8. Continuity of gluing maps and obstruction sections 68 9. Diffeomorphisms of moduli spaces defined by paths of metrics 88 References 106

1. Introduction

The underlying problem which motivates the present work and its sequels [14], [15], [16] is to prove the strong form of the Kotschick-Morgan conjecture for the Donaldson invariants of a four-manifold with b^{+} = 1 and b^{1} = 0, namely Conjecture 6.2.2 in [36]. From work of Göttsche we know that this (in fact, even the weaker Conjecture 6.2.1 of [36]) allows one to completely determine the wall-crossing formula for such four-manifolds. Though the two problems are not directly related, the computational and analytical problems arising in attempts to prove the wall-crossing formula are analogous, but still significantly simpler, than those appearing in our concurrent program to prove Witten’s conjecture [20], [17].

Our program to relate the Donaldson and Seiberg-Witten invariants [21], [20], [17], [18] requires a gluing theory, for PU(2) monopoles, possessing most of the properties of the gluing theory established here and in [14] for the easier case of anti-self-dual connections. Thus, we shall present the results we need in the considerably simpler case of anti-self-dual connections separately, so we can address elsewhere the difficulties peculiar to PU(2) monopoles [18]. The other purpose of the present article and its sequels is, of course, to prove the Kotschick-Morgan conjecture. That too bears on our program to prove the relation between Donaldson and Seiberg-Witten invariants, as it allows us to dispose of the technical difficulties in the general method underlying the expected proof of equivalence in an easier case. Our understanding of the work of Kronheimer and Mrowka is that similar difficulties — namely, possible contributions from lower-level ‘reducibles’ (Seiberg-Witten monopoles in the case of PU(2) monpoles) — afflict their program to relate the Yang-Mills and Seiberg-Witten Floer homologies (and thus prove property P for knots).

The principal difficulties arising in attempts to prove the Kotschick-Morgan conjecture are due to ‘bubbling’, a phenomenon which also accounts for the problems arising when one considers either PU(2) monopoles or pseudo-holomorphic curves in symplectic manifolds (of any dimension).

1.1. Statement of results. The main purpose of the present article and its companion [14] is to prove a general gluing theorem for anti-self-dual connections. Naturally, many statements of gluing theorems have appeared before in the literature [5], [8], [9], [49], [52], [54], [55], [56], so the reader may reasonably ask what is new and different about the present version. The gluing-theorem statements that are closest to those needed for a proof of the Kotschick-Morgan conjecture are given by Theorems 3.4.10 and 3.4.17 in [25]: both assertions are based on work of Taubes, principally [52], [54], [55], [56], though the assertions of [25] go beyond what has been already proved in or is easily adapted from the existing gluing-theory literature. In any event, the statement of the gluing theorem given in [25] is not sufficient to prove the Kotschick-Morgan conjecture; some of the additional requirements are briefly explained below in §1.3.

The present article is intended to bridge the gap between the gluing-theory literature cited above and two assertions of [25, Theorems 3.4.10 & 3.4.17], one implicit and one explicit, as we describe in §1.1.1 and §1.1.2 below.

1.1.1. Diffeomorphisms between moduli spaces of instantons over S^{4}. As is well known, the bundles Gl_{κ}^{w}(Σ, λ_{0}) of ‘gluing data’ [25, Eq. (3.4.9)], [25, pp. 250–253] — defining the domain of the gluing maps and determined by a choice of smooth stratum Σ ⊂ Sym^{ℓ}(X) and small positive constant λ_{0} (the maximum ‘scale’ size) — contain moduli spaces of ‘centered’ SU(2) connections on S^{4}, anti-self-dual with respect to its standard metric. Implicit in the statements of [25] is the assumption that Taubes’ perturbation scheme can be used to deform almost g-anti-self-dual connections over X, produced by splicing the gluing data, to nearby g-anti-self-dual connections in such a way that the resulting gluing map is a homeomorphism onto an open subset of M_{κ}^{w}(X, g). (We reserve the term ‘splicing’ for the process of cutting and pasting connections and the term ‘gluing’ for the composition of the splicing and the subsequent deformation to a family of anti-self-dual connections; see §3 for definitions of the bundles of gluing data.) However, in the situation considered in [25], the moduli spaces on S^{4} are allowed to have arbitrary instanton number and in this case, the Taubes’ gluing theory of [56] requires one to compose the splicing map with a certain homotopy of B_{κ}^{w,∗}(X) in order to produce a glued-up family of anti-self-dual connections. The difficulty — a significant one — is due to the discrepancy between the conformally flat metric on S^{4} and the non-flat metric on X. This means that the self-dual curvatures of the spliced connections will not satisfy the estimates required to perturb them to a family of anti-self-dual connections when the families of centered instantons over S^{4} being spliced are non-compact, which is always the case for the full, centered instanton moduli space when the instanton number is greater than one.

For example, suppose we attempted to use a Taubes deformation scheme to directly establish a diffeomorphism from M_{κ}(S^{4}, g_{0}) to M_{κ}(S^{4}, g), where g_{0} is the standard round metric on S^{4} and g is a nearby metric; typically, such metrics arise by pulling back the metric on X near a splicing point x_{0} via Gaussian coordinates, conformal dilation, and splicing with g_{0} to give a smooth, almost-round metric on S^{4} = (T X)_{x_{0}} ∪ {∞}. One might naively expect that one could construct a diffeomorphism by solving the equation

P_{+}(g)(d_{A}a + a ∧ a) = −P_{+}(g)F_{A}, (1.1)

for a deformation a ∈ L_{k}^{2}(S^{4}, Λ^{1} ⊗ 𝔤_{E}), yielding an SU(2) connection A + a over S^{4} with P_{+}(g)F_{A+a} = 0, for each SU(2) connection A over S^{4} with P_{+}(g_{0})F_{A} = 0. If the points [A] above are constrained to lie in a subset U ⊂ M_{κ}(S^{4}, g_{0}), then a solution to the anti-self-dual equation (1.1) is possible by the fixed-point method (as in [8], §7 in the present article, or [56]), provided we have bounds

\begin{aligned}sup_{[A]∈U} ∥P_{+}(g)F_{A}∥_{L^{p}(S^{4})} & ≤ M, \\ sup_{[A]∈U} ∥P_{+}(g)F_{A}∥_{L^{♯,2}(S^{4})} & ≤ ε, \end{aligned} (1.2)

where M = M(κ, g) < ∞ is a positive constant and ε = ε(κ, g) is a sufficiently small positive constant; the ‘critical exponent’ Sobolev L^{♯,2} norms are explained in [12]. The size of ε (and what it depends on) varies with the choice of norms used in the perturbation scheme, as does the precise choice of Sobolev norms: we would use p = ∞ in §7 (see our Proposition 7.6) while Taubes would use p = 4 in [56] (see his Lemma 5.1 and the surrounding discussion). The important point to note is that if U is a compact subset of M_{κ}(S^{4}, g_{0}), then we can easily achieve the bounds (1.2) just by requiring g to be suitably close to g_{0}: indeed, this is exactly the situation in [5], [8], [52], and [54], where the subsets U are indeed be assumed to be compact with respect to the usual quotient L_{k}^{2} topology (for example, the moduli space of centered 1-instantons on S^{4}, which is a single point). However, if we take U to be either the full moduli space of κ-instantons on S^{4} when κ ≥ 1 or the moduli space of centered k-instantons on S^{4} when κ ≥ 2, the subset U is non-compact and uniform bounds such as (1.2) are impossible to achieve.

Taubes partially circumvents this problem in [56, §5] by showing that one can still homotope a family of connections in B_{κ}^{w,∗}(X, g) with ∥F_{A}^{+}∥_{L^{2}} < ε to a family which does satisfy the required estimates. He can then perform the desired perturbation to a family of anti-self-dual connections; see [56, Lemma 5.1]. However, one no longer knows whether the resulting composite map is a diffeomorphism, or a homeomorphism from the closure of the gluing data. In [5], [8], [52], [54], the families of centered instantons on S^{4} are assumed to be compact (the problem then disappears) either by supposing that the instanton number is one or by picking a fixed connection on S^{4}, that is, a single point in the moduli space over S^{4}.

The first alternative to employing homotopies, as in [56], might be to simply allow the metrics on S^{4} to deviate slightly from the standard round metric of radius one, in such a way that the metrics near the north poles match the metric g on X where the instanton is to be spliced onto the background g-anti-self-dual connection on X. While this strategy would eliminate the problems with a priori estimates, the space of gluing data Gl_{κ}^{w}(Σ, λ_{0}) would not necessarily possess the fiber-bundle structure that seems to be a key ingredient in any attempt to prove the Kotschick-Morgan conjecture: the moduli spaces of instantons on S^{4} would vary with the multi-sets {x_{1}, …, x_{ℓ}} ⊂ Σ and it is not clear that the space Gl_{κ}^{w}(Σ, λ_{0}) would be a locally-trivial fiber bundle over Σ as assumed, say, in [25, Eq. (3.4.9)] and in related applications such as [40], [50], [62], [64]. In our own approach to the intersectiontheory problem, [15], [16], the existence of locally-trivial fiber-bundle structures plays a crucial role in the calculation of the wall-crossing terms. Furthermore, it is not clear how one could define differentials of the resulting gluing map without the benefit of local triviality.

The second alternative — the one we shall ultimately use — would be to try to show that moduli spaces of instantons on S^{4}, defined by the standard metric g_{0} and a nearby metric g′_{0} are indeed diffeomorphic, despite the aforementioned difficulties. In practice, this is a much more delicate problem than it might appear and it is not implied (again, because the a priori estimates on the self-dual curvature components are too weak) by standard perturbation schemes, such as those described in [8], [56]. The instanton moduli spaces can be viewed as fibers of the projection map π : 𝔐_{κ}(S^{4}) → Met(S^{4}), where 𝔐_{κ}(S^{4}) ⊂ B_{κ}(S^{4}) × Met(X) is the universal moduli space of anti-self-dual SU(2) connections on S^{4} with second Chern number κ. Because the projection π is not proper, it is not clear that the fibers π^{−1}(g_{0}) and π^{−1}(g′_{0}) are necessarily diffeomorphic, no matter how close the metrics g_{0} and g′_{0}: by abstract differential topology alone one can at most say that the fibers are cobordant. The difficulty is discussed further in our introduction to §9.

With the preceding remarks understood, we can state our two main results. The first bears on the basic technical question concerning the comparison of moduli spaces of antiself-dual connections defined by nearby metrics. In particular, in order for one to have a locally trivial cone bundle neighborhood structure for M̄_{κ}^{w}(X, g) (see [25] for definitions) one needs to know that the moduli space of anti-self-dual connections for metrics on S^{4} which are close to standard are in fact diffeomorphic to that defined by the standard metric of radius one. More precisely, we prove:

Theorem 1.1. There is a positive constant ε such that the following holds. Let g_{0} be the standard metric of radius one on the four-sphere S^{4}. Suppose γ : I → C ⊂ Met(S^{4}), t ↦ g_{t}, is a C^{∞} path of C^{∞} metrics on S^{4} through g_{0}, t ∈ I ⊂ ℝ, such that

sup_{t∈I} ∥g_{t} − g_{0}∥_{C^{1}(S^{4},g_{0})} < ε.

Let κ be a positive integer and let M_{κ}(S^{4}, C) := π^{−1}(C), where π : 𝔐_{κ}(S^{4}) → Met(S^{4}) is the projection from the universal moduli space 𝔐_{κ}(S^{4}) = {[A, g] ∈ B_{κ}(S^{4})×Met(S^{4}) : F_{A}^{+,g} = 0} onto Met(S^{4}). Then the following hold.

• There is a C^{∞} diffeomorphism

Ψ : M_{κ}(S^{4}, g_{0}) × I ≅ M_{κ}(S^{4}, C), ([A], t) ↦ Ψ([A], t),

such that (γ^{−1} ∘ π) ∘ Ψ = π_{2}, where π_{2} : π^{−1}(g_{0}) × I → I is projection onto the second factor and γ^{−1} is the diffeomorphism C → I, γ(t) ↦ t. In particular, the map Ψ restricts to a diffeomorphism

Ψ_{t} = Ψ(·, t) : M_{κ}(S^{4}, g_{0}) ≅ M_{κ}(S^{4}, g_{t}), for t ∈ I.

• The map Ψ_{t} = Ψ_{t}^{κ} uniquely extends to a smoothly stratified diffeomorphism Ψ̄_{t} : M̄_{κ}(S^{4}, g_{0}) → M̄_{κ}(S^{4}, g_{t}), that is, a homeomorphism of their Uhlenbeck compactifications, which restricts to a diffeomorphism on all smooth strata. The restriction of Ψ̄_{t} to a stratum M_{ℓ}(S^{4}, g_{0}) × Σ, where Σ ⊂ Sym^{ℓ}(X) and 0 < ℓ < κ, is given by the diffeomorphism

Ψ̄_{t} : M_{ℓ}(S^{4}, g_{0}) × Σ ≅ M_{ℓ}(S^{4}, g_{t}) × Σ, ([A], x) ↦ (Ψ_{t}^{ℓ}([A], g), x).

Remark 1.2. Although we shall make no use of generalizations of Theorem 1.1, the conclusions also hold for any oriented, smooth homotopy four-sphere X equipped with a generic Riemannian metric. More generally still, for any closed, connected, oriented, smooth fourmanifold X, the basic constraint on the data (X, g, w, κ) is that the Uhlenbeck compactification M̄_{κ}^{w}(X, g) contain no points [A, x] with non-trivial H_{A}^{2} and M_{ℓ}^{w}(X, g) contain no points with nontrivial H_{A}^{0} when 0 < ℓ ≤ κ and g is near g_{0}.

We shall keep the standard notational practices of [8], [38]; for completeness, we summarize the relevant conventions in §2.1.

1.1.2. Continuity of gluing maps. We now turn to our second principal result. Taubes’ gluing maps [56] are defined on the level of connections, rather than ideal connections, and it is not clear from [52], [54], [55], [56] that they necessarily extend continuously to the closure Ḡl_{κ}^{w,+}(X, Ū_{0}, Σ̄, λ_{0}) of the gluing data Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}). See §3.2 for the definitions of the gluing data and its closure defined by a choice of open neighborhood U_{0} in M_{κ_{0}}^{w}(X, C) of the ‘background’ connection [A_{0}], together with a smooth stratum Σ ⊂ Sym^{ℓ}(X) and small positive constant λ_{0}. Let

B̄_{κ}^{w}(X) := ⋃_{ℓ=0}^{⌊κ⌋} B_{κ−ℓ}^{w}(X) × Sym^{ℓ}(X),

where ⌊κ⌋ denotes the greatest integer less than or equal to κ. Our second main result is a proof of the continuity assertion in [25]:

Theorem 1.3. Let X be a closed, oriented C^{∞} manifold with generic C^{∞} Riemannian metric g. Let w ∈ H^{2}(X; ℤ) be a class such that no SO(3) bundle over X with second Stiefel-Whitney class equal to w (mod 2) admits a flat connection. Let E be a rank-two, Hermitian vector bundle over X with c_{1}(E) = w and c_{2}(E) − c_{1}(E)^{2}/4 = κ ≥ 1. Let 0 < ℓ < ⌊κ⌋ be an integer, let Σ ⊂ Sym^{ℓ}(X) be a smooth stratum, let C ⊂ GL(T^{∗}X) be a compact, connected, smooth submanifold-with-boundary containing the identity, let g ∈ Met(X), and let [A_{0}] ∈ M_{κ−ℓ}^{w}(X, g). Then there are

• A smooth gluing map γ : Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) → B_{κ}^{w,∗}(X) which extends to a continuous map γ : Ḡl_{κ}^{w,+}(X, Ū_{0}, Σ̄, λ_{0}) → B̄_{κ}^{w}(X). Here, λ_{0} is a sufficiently small positive constant, depending at most on X, g, κ.

• A smooth Kuranishi obstruction section χ of a smooth Kuranishi obstruction vector bundle γ^{∗}𝔙 over Gl_{κ}^{w,+}(X, U_{0}, Σ, λ), where 𝔙 is a continuous vector bundle over an open neighborhood U ⊂ B̄_{κ}^{w}(X)×C of [A_{0}]×Sym^{ℓ}(X) which restricts to a smooth bundle over U ∩ (B_{κ}^{w,∗}(X) × C). The bundle 𝔙 has real rank equal to the real dimension of H_{A_{0}}^{2}. The section χ is the restriction of a continuous section on U.

Together, the gluing map and obstruction section have the property that

γ(χ^{−1}(0)) ⊂ M̄_{κ}^{w}(X, C) ∩ U,

where M̄_{κ}^{w}(X, C) is the Uhlenbeck compactification of the universal moduli space M_{κ}^{w}(X, C) of connections on 𝔰𝔲(E), anti-self-dual with respect to the family of metrics {f(g) : f ∈ C} ⊂ Met(X).

The gluing maps and obstruction sections are defined in §3 and §7.

Remark 1.4. In the statements of the gluing theorem given in [25] (see Theorems 3.4.10, 3.4.17, and 3.4.18), it is assumed that the gluing data bundle Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) → Σ is replaced by its restriction Gl_{κ}^{w}(X, U_{0}, K, λ_{0}) → K, where K ⋐ Σ is a precompact open subset. We make no such restriction here, so the points x ∈ Σ are allowed to approach the diagonals of the symmetric product Sym^{ℓ}(X) containing the non-compact smooth stratum Σ.

There are several important properties which are required of a useful gluing theory — in any context, but particularly for applications to the Kotschick-Morgan conjecture — which are not asserted by Theorem 1.3. These extensions are discussed in §1.3 and are derived in the companion article [14].

We shall need the case where C is the image of a path γ of metrics on T^{∗}X and the connection A_{0} is a non-flat reducible, with stabilizer S^{1}. In this instance, an open neighborhood U ∩M_{κ}^{w}(X, γ) of [A_{0}]×Sym^{ℓ}(X) will be covered by a collection of images of gluing maps, one gluing map per smooth stratum Σ of Sym^{ℓ}(X).

To satisfy the constraint on the second Stiefel-Whitney class, we use the now standard trick of Morgan and Mrowka of requiring the existence of a spherical class β ∈ H_{2}(X; ℤ) such that ⟨c_{1}(E), β⟩ ≠ 0 (mod 2). This condition can always be satisfied by blowing up if necessary and so, as far as the computation of Donaldson invariants is concerned, there is no loss in generality, but the structure of the moduli space is considerably simpler as flat connections are excluded from the Uhlenbeck compactification M̄_{κ}^{w}(X, g). For generic paths of metrics, γ, most, but not all, non-flat reducibles occurring in M̄_{κ}^{w}(X, γ) can be assumed to have vanishing obstruction spaces, so this explains the appearance of the obstruction map in the statement of Theorem 1.3. After work on the continuity assertions of Theorem 1.3 had been written up, we learned that Graham Taylor [57] had considered the issue of continuity, in a more restricted form, of the gluing maps defined by Taubes in [54].

1.2. Outline. We outline the contents of the remainder of the article. The Kotschick-Morgan conjecture and how its solution bears on the expected solution to Witten’s conjecture are explained in §2.

In §3 we define the ‘splicing map’ for connections via the usual cut and paste techniques. Of course, this part is entirely standard: the main new issue is that for later purposes it is necessary to keep track of the constants defining the splicing map itself. The splicing maps yield continuous maps (smooth away from singularities) from bundles of gluing data, one bundle per smooth stratum Σ ⊂ Sym^{ℓ}(X), into the quotient space B_{κ}^{w,∗}(X) whose image lies suitably close to the stratum M_{κ−ℓ}^{w}(g)×Σ of M̄_{κ}^{w}(g). The question of exactly how ‘close’ is addressed in §4 and §5.

In §4 we record the regularity theory for the anti-self-dual equation. The exposition is modeled on the corresponding theory for the PU(2) monopole equations and, indeed, applies to any quasi-linear first-order elliptic equation with a quadratic non-linearity on a fourmanifold. The results are used repeatedly throughout the present article and its sequels, so precise statements are given. The general pattern is that one typically has uniform, global L_{1}^{2} estimates for solutions to the anti-self-dual equation (obtained, for example, by deforming an approximate, spliced solution), but when we need estimates with respect to stronger norms — uniform with respect to a non-compact parameter — we must get by with only local estimates.

We shall almost always want estimates — for spliced connections or their curvatures — whose constants depend at most on the L^{2} norms of the curvatures, or the L^{∞} norms of the self-dual curvatures, of the component connections restricted to suitable open sets. There are different ways these estimates can be achieved, depending on the geometry of the region in X where estimates are required. One option is to use decay estimates, due in varying degrees of generality to Donaldson, Groisser-Parker, Morgan-Mrowka-Ruberman, or Råde. However, since our real interest is in the case of PU(2) monopoles, we provide in §5 a slightly simpler approach here which avoids the use of such decay estimates or their generalizations.

This section concludes with an estimate for the self-dual component of the curvature of a spliced connection; as this is a typical application of decay-type estimates for anti-self-dual connection we present the details in this case and then leave subsequent calculations of this type for the reader.

Splicing anti-self-dual connections produces a family, A, of approximately anti-self-dual connections on X and thus a family of Laplacians, d_{A}^{+}d_{A}^{+,∗}, on L^{2}(X, Λ^{+} ⊗ 𝔰𝔲(E)). We shall need upper and lower bounds for the small eigenvalues of this Laplacian, along with estimates for its eigenfunctions and variations with respect to A. These and related estimates are derived in §6, the methods and results being modeled on those of Taubes [52], [56].

In §7 we prove the existence and regularity of solutions to the anti-self-dual equation, the essential point being to provide estimates for the solutions using a slightly stronger system of norms than those of [52], [56]: the availability of slightly stronger estimates is important in [14]. Our basic requirement on the Hermitian bundle E is that no SO(3) bundle with second Stiefel-Whitney class equal to c_{1}(E) (mod 2) admits a flat connection; we do not require that X be simply-connected or even that b^{1}(X) = 0. Taking w = c_{1}(E), this condition ensures that the Uhlenbeck compactification M̄_{κ}^{w}(X, g) contains no flat connections and, in particular, no connections A_{0} with non-vanishing H_{A_{0}}^{2} except reducibles.

In §8 and §9 we are finally in a position to prove the two main results of the present article. The first, which is the subject of §8, concerns the continuity of the gluing map with respect to Uhlenbeck limits: in general, we need to know that if either a proper subset of the scales (used to attach the connections from S^{4}) tends to zero or if the SU(2) connections on S^{4} converge to an Uhlenbeck limit, then the sequence of families of glued-up solutions converge to a family of ideal glued-up solutions. For splicing maps, this property is an immediate consequence of the definition, which is purely local, but for gluing maps it is not obvious because of the global dependence of the solutions on the gluing data via Green’s operators. The only continuity statement that is a direct consequence of Taubes’ original construction is that if all the scales tend to zero then the glued-up solutions converge to the obvious ideal points in the Uhlenbeck compactification.

The proof of our second main result is completed in §9. If one chooses the standard metric on S^{4} and splices in the usual way, then one will obtain a family of approximately anti-self-dual connections which only have uniformly bounded C^{0} norms of F_{A}^{+,g} when the metric g on X is (conformally) flat near the gluing sites. It is easy to see that in general, when the metric g on X is not flat near the gluing sites, one will at best have a uniform L^{2} bound on F_{A}^{+,g}: this is not sufficient for the fixed-point argument to work, whether one uses the scheme employed here in §7 or the similar one given in [56]. The problem is partly circumvented in [56] by showing that a certain gradient flow (see [53]) yields a homotopy from the image of the splicing map into a neighborhood in B_{κ}^{w}(X, g) whose connections satisfy the required a priori estimates on F_{A}^{+,g}. While homotopies suffice for Taubes’ applications to the stable Atiyah-Jones conjecture in [56], they do not suffice for our construction of links. Instead, we construct a diffeomorphism between the moduli spaces M_{κ}(S^{4}, g_{0}) and M_{κ}(S^{4}, g′_{0}), where g_{0} is the standard round metric of radius one and g′_{0} is a ‘nearby’ metric which varies with the location of the gluing site x ∈ X. That such a diffeomorphism should exist is less obvious than it might appear, even when the metrics are suitably close, because the spaces are non-compact. Despite this difficulty, we show that a modified version of Taubes’ flow does yield a diffeomorphism and this completes the proof of Theorem 1.1.

1.3. Sequels. Some remarks are in order concerning the contents of the companion articles [14], [15], [16].

1.3.1. Surjectivity of gluing maps. The reader will note that the main result proved here is less than the ‘first half’ of a desired ‘gluing theorem’. To be of use for parameterizing neighborhoods of ideal reducibles, we would also need to know the following:

1. The gluing map γ is a smooth embedding of Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) and a topological embedding of the Uhlenbeck closure of the gluing data, Ḡl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}).

2. The image of Ḡl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) ∩ χ^{−1}(0) under γ is an open subset of M̄_{κ}^{w}(X, C);

3. The space M̄_{κ}^{w}(X, C) has a finite covering by such open subsets. These remaining properties (1)–(3) are proved in [14], together with certain C^{1} differentiability properties of γ, and comprise the ‘second half’ of the gluing theorem. It is important to note that preceding three gluing-map properties are not simple consequences of the gluing method itself and their justification constitutes the more difficult half of the proof of the full gluing theorem. In particular, despite an extensive literature on gluing theory for anti-self-dual connections, the existing accounts (see, for example, [5], [8], [49], [52], [54], [55], [56]) only address very special cases which do not capture all of the difficulties one encounters when attempting to solve the general problem.

1.3.2. Bubble-tree compactifications and manifolds-with-corners structures. The proof of injectivity and surjectivity of the gluing maps relies on estimates for the differentials of the gluing maps: these estimates in turn yield further information concerning the C^{1} differentiable structure of the bubble-tree compactification of M_{κ}^{w}(X, g). Indeed, related work of Taubes in [51] suggests that the bubble-tree compactification should be a C^{1} manifold with corners (as we would anticipate from known results concerning the moduli space of stable maps on a symplectic manifold — an orbifold). In our case, the C^{1} differentiability properties of the gluing maps essentially imply that the bubble-tree compactification of M_{κ}^{w}(X, C) can be given the structure of a C^{1} manifold with corners [15].

1.3.3. Intersection theory and completion of the proof of the Kotschick-Morgan conjecture. In work in preparation, [16], we apply the preceding gluing results to compute the wallcrossing contribution of ideal reducibles in a one-parameter family of anti-self-dual moduli spaces, M̄_{κ}^{w}(X, g_{t}), t ∈ [0, 1], to the change in the Donaldson invariants of X. Specifically, we show that the changes only depend on the homotopy type of X. The importance of the continuity assertion for gluing maps and that some condition on the differentials of the gluing maps would be needed in order to compare overlaps of different gluing-map images — with applications to a proof of the Kotschick-Morgan conjecture — first arose in discussions with John Morgan and Peter Ozsváth on stratified spaces [43].

1.4. Acknowledgments. The first author is grateful for the hospitality and support of the many institutions where various parts of the present article and its sequels where written, namely Harvard University, beginning in Autumn 1993, Ohio State University (Columbus), the Max Planck Institut für Mathematik (Bonn) — where the principal new parts of the article were written — together with the Institute des Hautes Etudes Scientifiques (Buressur-Yvette) and the Institute for Advanced Study (Princeton), where the article was finally completed. He would also like to thank the National Science Foundation for their support during the preparation of this article.

2. The wall-crossing formula for Donaldson invariants

Our first task is to give a precise statement of the conjectured shape of the general wallcrossing formula. There are two slightly different versions given in [36]: the one we aim to prove is a stronger form of the second conjecture. It is only fair to admit straightaway that since the advent of the Seiberg-Witten invariants, the conjecture itself no longer appears to have a direct bearing on smooth four-manifold topology, though it is still an important aspect of Yang-Mills gauge theory. Nevertheless, the conjecture does retain considerable mathematical interest or, more precisely, its method of proof because this is expected to provide a simpler illustration of the proof of the corresponding conjecture for PU(2) monopoles [20].

Our second task in this subsection is to describe the Kuranishi models for essential singularities in parameterized moduli spaces M_{κ}^{w}(X, γ) when b^{+}(X) = 1 and both the class w ∈ H^{2}(X; ℤ) and path of metrics γ : [−1, 1] → Met(X) have been chosen so as to eliminate all singularities except those due to non-flat reducible connections at a critical metric g_{0} = γ(0). The properties of the model in turn dictate our approach to gluing theory and the construction of models for singularities due to reducibles occurring in all strata of the compactification M̄_{κ}^{w}(X, γ).

Reducible connections in M_{κ}^{w}(X, g_{0}) occur in (possibly obstructed) families of dimension b^{1}(X). Links of positive-dimensional families of singularities in moduli spaces of PU(2) monopoles are constructed in [17] but, as might be expected, they are considerably more complicated than those of isolated singularities. Neighborhoods of isolated reducibles in lower strata of M̄_{κ}^{w}(X, γ) may be parameterized via Taubes’ approach to gluing, which uses spectral projections to construct an equivariant, finite-rank normal bundle of the singularity in a thickened moduli space which contains the singularity as a smooth submanifold. This is no longer possible in the case of positive-dimensional families, due to spectral flow, and one must use a combination of the link construction in [17] and a gluing method which leans closer to that of Donaldson [5], [8]. Gluing constructions of the latter kind lead to further difficulties: see [18], [19]. However, the formulas for the wall-crossing terms have only been predicted or proved modulo the Kotschick-Morgan conjecture, when b^{1}(X) = 0. Hence, for these reasons, we shall restrict our attention to this case whenever necessary to simplify the discussion at hand.

2.1. Preliminaries. To avoid ambiguity, we gather here the notational conventions we shall employ throughout our work; we follow, whenever possible, the usual practices of [8],

[38] and, in a few instances, those of [25].

Let E be a Hermitian, rank-two vector bundle over S^{4}. Let A_{κ} = A_{E} = A_{E}(S^{4}) be the space of L_{k}^{2} connections on the SO(3) bundle 𝔤_{E} = 𝔰𝔲(E) over S^{4}. Let G_{E} be the group of L_{k+1}^{2} unitary gauge transformations of E with determinant one and let G_{E}^{s} ⊂ G_{E} be the subgroup consisting of gauge transformations u ∈ G_{E} such that u(s) = id_{E|_{s}}.

Let B_{κ}^{s} = B_{E}^{s} = B_{E}^{s}(S^{4}) be the G_{E}^{s}-quotient space of L_{k}^{2} connections on the bundle 𝔤_{E} over S^{4}. There is a map, defined by the choice q ∈ Fr(𝔤_{E})|_{s},

B_{E}^{s} ≃ A_{E} ×_{G_{E}} Fr(𝔤_{E})|_{s}, [A] ↦ [A, q],

which is easily seen to be a homeomorphism. The existence of a slice for the action of the based gauge group, G_{E}^{s}, has been established by Groisser and Parker and it then follows that this homeomorphism is in fact a diffeomorphism [30], as there are no non-flat reducible SO(3) connections over S^{4}. The group SO(3) = SU(2)/{±id} acts freely on B_{E}^{s} by the assignment [A] ↦ [uAu^{−1}] or [A, q] ↦ [A, qu], if u ∈ SU(2).

Let Fr(𝔤_{E}) denote the principal SO(3) bundle of oriented, orthonormal frames for 𝔤_{E}. Let Fr(T X) denote the principal SO(4) bundle of oriented, g-orthonormal frames for T X.

2.2. The Kotschick-Morgan conjecture. Let X be a closed, oriented, smooth fourmanifold with b^{+}(X) = 1. We briefly recall the definition of Donaldson invariants in this case [6], [26], [27], [28], [35], [36]. Following [26, §1], we call

Ω_{X} := {β ∈ H^{2}(X; ℝ) : β^{2} > 0}/ℝ^{+}

the positive cone of H^{2}(X; ℝ)/ℝ^{+}, where we denote ℝ^{+} = (0, ∞). Fix a class w ∈ H^{2}(X; ℤ) and a number κ ∈ \frac{1}{4}ℤ such that

−4κ = w^{2} (mod ℤ) and 8κ − 3(2 − b^{1}) ≥ 0. (2.1)

A wall of type (w, κ) is a nonempty hyperplane in Ω_{X},

W^{ξ} := {β ∈ Ω_{X} : β · ξ = 0} = ξ^{⊥} ∩ Ω_{X}, (2.2)

for some class ξ ∈ H^{2}(X; ℤ) with

ξ = w (mod 2) and p ≤ ξ^{2} < 0, (2.3)

where p = −4κ. Let ℓ = (ξ^{2} − p)/4: observe that 0 ≤ ℓ < κ and that ℓ is an integer. ^{1} According to [6] the set of (w, κ)-walls is locally finite.

The connected components of the complement in Ω_{X} of the walls of type (w, κ) are called the chambers of type (w, κ). Require that w (mod 2) be such that no SO(3) bundle V over X with w_{2}(V) = w (mod 2) admits a flat connection; see [35], [46] for sufficient criteria and how they may be achieved with no loss of generality [46] for the purposes of computing Donaldson invariants. Then the Donaldson invariants of X associated to (w, κ) are defined with respect to chambers of type (w, κ). We shall write

D_{X,C}^{w}(z), z ∈ 𝔸(X),

for the Donaldson invariants defined by the class w ∈ H^{2}(X; ℤ) and the chamber C. Indeed, for a Riemannian metric g on X, let ω ∈ Ω^{+,g}(X) be a g-self-dual, harmonic two-form (so d^{+,g}d^{∗,g}ω = 0) such that ∫_{X} ω ∧ ω = 1. The form is unique up to sign, so fix an ω and let it define an orientation for H^{+}(X; ℝ). Let (−1) ∧ ω ∈ det(H^{0}(X; ℝ) ⊕ H^{+}(X; ℝ)) define the choice of homology orientation. For generic g, the point [ω] ∈ H^{2}(X; ℝ) does not lie on a wall W. If C ⊂ H^{2}(X; ℝ)/ℝ^{+} is the chamber containing [ω] then, according to [35, Theorem 3.2], [36, Theorem 3.0.1], one has D_{X,g}^{w}(z) = D_{X,C}^{w}(z), that is, the Donaldson invariants only depend on the chamber C containing the period point [ω(g)] and not the metric g.

1The case ξ^{2} = 0 and ℓ = κ could occur if the background reducible connection were flat. However, this possibility is excluded by choosing w as suggested in [46]; see Lemma 2.4.

Recall that M_{κ}^{w}(X, g) has the expected dimension

dim M_{κ}^{w}(X, g) = −2w^{2} − 3(b^{+} − b_{1} + 1)
= −2w^{2} − 3(2 − b_{1}).

If deg z ≠ −2w^{2} − 3(2 − b_{1}) (mod 8) then D_{X,C}^{w}(z) = 0. Otherwise, find κ ∈ \frac{1}{4}ℤ such that

deg z = 8κ − 3(2 − b_{1})

and a Hermitian, rank-two vector bundle E over X such that

c_{1}(E) = w and − \frac{1}{4}p_{1}(𝔤_{E}) = ⟨c_{2}(E) − c_{1}(E)^{2}/4, [X]⟩ = κ. (2.4)

where, throughout this paper, we denote 𝔤_{E} := 𝔰𝔲(E). We now define D_{X,C}^{w}(z) following the recipe of [38, §2].

Suppose that C_{+} and C_{−} are separated by a single wall W^{ξ}; there may be more than one class ξ of type (w, κ) defining W^{ξ}. The difference

δ_{X,C_{+},C_{−}}^{w}(z) := D_{X,C_{+}}^{w}(z) − D_{X,C_{−}}^{w}(z), z ∈ 𝔸(X),

defines a transition formula. Based on known examples and their own work, Kotschick and Morgan proposed the following two conjectures:

Conjecture 2.1. The transition formula δ_{X,C_{+},C_{−}}^{w} is a homotopy invariant of the pair (X, ξ); more precisely, if ϕ is an oriented homotopy equivalence from X′ to X, then

δ_{X,ϕ^{∗}(C_{+}),ϕ^{∗}(C_{−})}^{ϕ^{∗}w} = ϕ^{∗}δ_{X,C_{+},C_{−}}^{w}.

This is expected to be equivalent to the following apparently stronger claim:

Conjecture 2.2. The transition formula δ_{X,C_{+},C_{−}}^{w} is a polynomial in ξ and the quadratic form Q_{X}, with coefficients depending only on ξ^{2}, homotopy invariants of X, and universal constants.

The property of the transition formula which we aim to establish is, in fact, stronger than that predicted by Conjecture 2.2. It is a remarkable result of Göttsche that if one assumes Conjecture 2.1 then the difference terms δ_{X,C_{+},C_{−}}^{w} can be computed explicitly for any simply-connected four-manifold. It seems very likely that the condition π_{1}(X) = 1 can be weakened to the case b^{1}(X) = 0, as Göttsche points out in [27] that he uses this constraint merely to remain consistent with the stated version of the Kotschick-Morgan conjecture in [36] and the Fintushel-Stern blow-up formula [23]: see [27, Corollary 5.1]. Göttsche’s argument only requires Conjecture 2.1 in order to replace X by ℂℙ^{1} × ℂℙ^{1} in [27, Lemma 4.10].

The first result in this direction is due to Donaldson, who gave a formula in the case w = 0 and κ = 1; these were extended by Kotschick in [35]. Using a gauge-theoretic approach, Yang [64] completely determined the transition formulas for w = 0 and ℓ = 1. This was further extended by Leness to the case ℓ = 2 and arbitrary w [40].

For a rational ruled complex surface X, the transition formulas for w = 0 and 2 ≤ κ ≤ 4 were determined by [41], [42], [58] using methods of algebraic geometry. Using Thaddeus’ method, Friedman and Qin completely determined the transition formulas when X is a rational surface with −K_{X} effective [26, p. 13]. Similar results were independently obtained by Ellingsrud and Göttsche [11], [10].

In a quite different direction, Göttsche [27] assumed the validity of the Kotschick-Morgan conjecture and used this assumption to explicitly compute the transition formulas, for arbitrary closed, oriented, smooth four-manifolds X with b^{1}(X) = 0 and b^{+}(X) = 1 in terms of modular forms.

2.3. Uhlenbeck compactness for parameterized moduli spaces. We recall the Uhlenbeck compactness results [8], [24] for the moduli space of anti-self-dual SO(3) connections and their natural extension to the case of families of metrics. Recall that the process of gluing up connections is essentially the inverse of Uhlenbeck degeneration of connections.

Definition 2.3. We say that a sequence of points [A_{α}, g_{α}] in B_{κ}^{w}(X) × Met(X) converges to a point [A, g, x] in B_{κ−ℓ}^{w}(X) × Met(X) × Sym^{ℓ}(X), where 0 ≤ ℓ ≤ κ is an integer, if the following hold:

• The sequence of metrics g_{α} converges in C^{r} to a metric g on T^{∗}X.

• Let E, E_{0} be rank-two Hermitian bundles over X, with determinant line bundle w and induced SO(3) bundles, 𝔤_{E} and 𝔤_{E_{0}}, with first Pontrjagin numbers −4κ and −4(κ−ℓ), respectively. There is a sequence of L_{k+1,loc}^{2}(X \ x), determinant-one, unitary bundle isomorphisms u_{α} : E|_{X\x} → E_{0}|_{X\x} such that the sequence of connections u_{α}(A_{α}) on 𝔤_{E_{0}}|_{X\x} converges to A on 𝔤_{E_{0}}|_{X\x} in L_{k,loc}^{2} over X \ x, and

• The sequence of measures |F_{A_{α}}|^{2} converges in the weak-* topology on measures to |F_{A}|^{2} + 8π^{2} ∑_{x∈x} δ(x), where δ(x) is the unit Dirac measure centered at the point x ∈ X.

Let C ⊂ Met(X) denote a compact subset of the Banach space of C^{r} Riemannian metrics on T^{∗}X (r ≥ 3) and define the corresponding parameterized moduli space by

M_{κ}^{w}(X, C) := ⋃_{g∈C} M_{κ}^{w}(X, g) × {g} ⊂ B_{κ}^{w}(X) × Met(X).

Let M̄_{κ}^{w}(X, C) be the closure of M_{κ}^{w}(X, C) in the space ∪_{ℓ=0}^{⌊κ⌋}M_{κ−ℓ} × Sym^{ℓ}(X) with respect to the induced Uhlenbeck topology, where ⌊κ⌋ is the greatest integer contained in κ(E) := −\frac{1}{4}p_{1}(𝔤_{E}), where −\frac{1}{4}p_{1}(𝔤_{E}) = ⟨c_{2}(E)− \frac{1}{4}c_{1}(E)^{2}, [X]⟩. We call the intersection of M̄_{κ}^{w}(X, C) with M_{κ−ℓ}^{w} × Sym^{ℓ}(X) a lower-level of the compactification M̄_{κ}^{w}(X, C) if ℓ > 0 and call M_{κ}^{w}(X, C) the top or highest level.

According to [8, Theorem 4.4.3 & Proposition 9.1.2], M̄_{κ}^{w}(X, C) is a compact, Haussdorf, second-countable space when endowed with the induced Uhlenbeck topology. We call the intersection of M̄_{κ}^{w}(X, C) with M_{κ−ℓ}^{w}(X, C)×Sym^{ℓ}(X) a lower-level of the compactification M̄_{κ}^{w}(X, C) if ℓ > 0 and call M_{κ}^{w}(X, C) the top or highest level: in general, the spaces M_{κ−ℓ}^{w}(X, C) and Sym^{ℓ}(X) are at best smoothly stratified.

The following lemma gives a simple condition on w which guarantees that there will be no flat SO(3) connections in M_{κ}^{w}(X, g), for any g. Because it relies only on w (mod 2), the lemma applies simultaneously to all levels of the Uhlenbeck compactification M̄_{κ}^{w}(X, g). ^{2}

Lemma 2.4. [46, p. 226] Let X be a closed, oriented four-manifold. Then • If E is a U(2) bundle over X and β ∈ H_{2}(X; ℤ) is a spherical class such that

2According to [22, p. 536], [35, Proposition 4.1], if V is an SO(3) bundle over a closed four-manifold X such that w_{2}(V) is not pulled back from H^{2}(K(π_{1}(X), 1); ℤ/2ℤ), then there are no flat connections on V.

However, one sees more directly that the Morgan-Mrowka criterion is satisfied via blow-ups.

⟨c_{1}(E), β⟩ ≠ 0 (mod 2), then no SO(3) bundle V over X with w_{2}(V) = c_{1}(E) (mod 2) admits a flat connection.

• In particular, if E is any U(2) bundle over X and Ê is the U(2) bundle over X̂ = X#\overline{ℂℙ}^{2} with c_{2}(Ê) = c_{2}(E) and c_{1}(Ê) = c_{1}(E) + e^{∗}, where e^{∗} = PD[e] and e = [\overline{ℂℙ}^{1}] ∈ H_{2}(X; ℤ), then no SO(3) bundle V over X̂ with w_{2}(V) = w_{2}(𝔰𝔲(Ê)) admits a flat connection. ^{3}

Proof. Let S ⊂ X be an embedded two-sphere representing β. If A were a flat SO(3) connection on V, then its restriction A|_{S} would be flat and thus a trivial connection on a trivial SO(3) bundle V |_{S}, as S is simply connected. But ⟨w_{2}(V |_{S}), [S]⟩ = ⟨c_{1}(E), β⟩ ≠ 0 (mod 2), yielding a contradiction. This proves the first assertion.

Replacing X by X̂ = X#\overline{ℂℙ}^{2} and E by Ê with c_{2}(Ê) = c_{2}(E) and c_{1}(Ê) = c_{1}(E) + e^{∗} ensures that ⟨c_{1}(Ê), e⟩ ≠ 0 (mod 2), as c_{1}(E)·e^{∗} = 0 and e^{∗} ·e^{∗} = −1. The second assertion follows.

2.4. Transversality for one-parameter families of moduli spaces. Rather than consider a moduli space of anti-self-dual connections defined by a single Riemannian metric, g_{0}, we shall need to consider C^{∞} paths of metrics γ : [−1, 1] → Met(X), where Met(X) is the Banach manifold of C^{r} metrics on T^{∗}X (with r ∈ ℤ large). For this purpose, we shall need the following basic transversality result [8], [24], [38]. Theorem 2.5. Let X be a closed, oriented, smooth four-manifold and let r ≥ 3 be an integer. Let v ∈ H^{2}(X; ℤ/2ℤ) be a class such that no SO(3) bundle with second Stiefel-Whitney class v admits a flat connection. Then the following hold:

• There is a first-category subset of the space of C^{r} metrics such that for all metrics g in the complement of this subset, the following hold: For any κ ∈ \frac{1}{4}ℤ and w ∈ H^{2}(X; ℤ) with −4κ = w^{2} (mod ℤ), the moduli space M_{κ}^{w,∗}(X, g) of irreducible g-anti-self-dual SO(3) connections is regular (as the zero-locus of the section F^{+,g}( ·) over B_{κ}^{w,∗}(X)) and thus a C^{∞} manifold of the expected dimension 8κ − 3(1 − b^{1} + b^{+}).

• When b^{+}(X) > 0, there is a first-category subset of the space of C^{r} metrics such that for all metrics g in the complement of this subset, the following hold: For any κ ∈ \frac{1}{4}ℤ and w ∈ H^{2}(X; ℤ) with −4κ = w^{2} (mod ℤ), the moduli space M_{κ}^{w}(X, g) of g-anti-self-dual SO(3) connections contains no reducible connections.

• Let g_{−1}, g_{1} be metrics satisfying the conditions of the preceding assertion and suppose b^{+}(X) > 0. Then there is a first-category subset of the space of C^{r} paths γ : [−1, 1] → Met(X) with γ(±1) = g_{±1} such that for all paths γ in the complement of this subset, the following hold. For any κ ∈ \frac{1}{4}ℤ and w ∈ H^{2}(X; ℤ) with −4κ = w^{2} (mod ℤ), the parameterized moduli space of γ-anti-self-dual SO(3) connections,

M_{κ}^{w,∗}(X, γ) := {([A], t) ∈ B_{κ}^{w,∗}(X) × [−1, 1] : F^{γ(t)}(A) = 0},

is regular (as the zero-locus of the section F^{+,γ(·)}( ·) over B_{κ}^{w,∗}(X)) × [−1, 1] and thus a C^{∞} manifold of the expected dimension. In particular, M_{κ}^{w,∗}(X, γ) gives a C^{∞} cobordism between the manifolds M_{κ}^{w,∗}(X, g_{−1}) and M_{κ}^{w,∗}(X, g_{1}).

3Because the Donaldson invariants of X#\overline{ℂℙ}^{2} determine and are determined by those of X, (see [23], [46]), no information about these invariants is lost by blowing up.

Proof. All of the above assertions are standard, so we merely list their provenance. The first assertion is proved as Corollary 4.3.18 in [8] and Theorem 3.17 in [24]: the manifold X is not required to be simply connected, as assumed in [8, §4.3], [24, §3], because flat connections are excluded by the condition on w and twisted reducible connections are smooth points by [38, Corollary 2.5]. Similarly, the third assertion follows from [8, Corollary 4.3.18], [38, Corollary 2.5], and the fact that flat connections are excluded. The second assertion is proved as Corollary 4.3.15 in [8]: the condition that X be simply connected is not required.

2.5. Kuranishi models and reducible anti-self-dual connections. It remains to consider the non-flat reducible connections in M̄_{κ}^{w}(X, γ), the only possible singular points of this compact cobordism which cannot be excluded by perturbations or a suitable choice of w (mod 2) or by dimension-counting when considering the intersection of geometric representatives with M̄_{κ}^{w}(X, γ). In this subsection we examine the elliptic deformation complex for a reducible [A_{0}] in M_{κ}^{w}(X, g_{0}), while in the following subsection we describe the Kuranishi model for a neighborhood of a reducible [A_{0}, g_{0}] in M_{κ}^{w}(X, γ).

In this section we classify the projectively anti-self-dual connections on E with non-trivial stabilizers in G_{E}. If A is a connection on E, we write A^{ad} for the connection induced by A on the associated SO(3) bundle 𝔤_{E} and denote its stabilizer by Stab_{A} ⊂ G_{E}. Except when discussing bundle splittings and reducible connections, our usual convention is to consider connections on 𝔤_{E} only, with their lifts to E via the fixed connection on det E being implicit when needed. We call A irreducible if Stab_{A} = {±1}, the center of SU(2), and reducible otherwise. The following classification is standard; see, for example, Proposition 3.1 in [17].

Lemma 2.6. Let X be a closed, oriented, smooth four-manifold with Riemannian metric g. Let E be a Hermitian two-plane bundle over X and suppose a unitary connection A on E represents a point [A] ∈ M_{E}. Then one of the following holds:

1. The connection A is irreducible, with stabilizer Stab_{A} = {±1}.

2. The connection A is reducible, but not projectively flat: the bundle E splits as an ordered sum of line bundles, E = L_{1} ⊕ L_{2}, the connection A has stabilizer Stab_{A} ≅ S^{1} and A = A_{1} ⊕ A_{2}, where A_{1} is a unitary connection on L_{1} and A_{2} = A_{e} ⊗ A_{1}^{∗} is the corresponding connection on L_{2} = (det E) ⊗ L_{1}^{∗}, with A_{e} being the fixed connection on det E.

3. The connection A is projectively flat and reducible with respect to a splitting E = L_{1} ⊕ L_{2}, where L_{2} = (det E) ⊗ L_{1}^{∗} and c_{1}(E) − 2c_{1}(L_{1}) ∈ H^{2}(X; ℤ) is a torsion class.

For the second case in Lemma 2.6. we remark that a projectively g-anti-self-dual connection A on E is reducible with respect to the splitting E ≅ L_{1} ⊕ L_{2} if and only if the induced g-anti-self-dual connection A^{ad} on 𝔤_{E} is reducible with respect to the splitting 𝔤_{E} ≅ iℝ ⊕ ξ, where ξ ≅ det E ⊗ L_{1}^{−2}. Indeed, we have

A = (\begin{matrix}A_{1} & 0 \\ 0 & A_{e} ⊗ A_{1}^{∗} \end{matrix}) and A^{ad} = 0 ⊕ A_{e} ⊗ 2A_{1}^{∗},

with F^{+,g}(A_{e} ⊗ 2A_{1}^{∗}) = 0.

From Theorem 2.5 we see that it remains to consider the case of reducible connections in M̄_{κ}^{w}(X, γ). Suppose that a class ξ ∈ H^{2}(X; ℤ) defines a (w, κ)-wall as in definition (2.2) so ξ = w (mod 2) and −4κ ≤ ξ^{2} < 0 as in (2.3). Let 0 ≤ ℓ < κ be the integer defined by −4(κ − ℓ) = ξ^{2}. Then there is a U(2) bundle E and a splitting E = L ⊕ w ⊗ L^{−1} with ξ = w ⊗ L^{−2}, where c_{1}(E) = w and p_{1}(𝔤_{E}) = −4(κ − ℓ) = ξ^{2}. The corresponding SO(3) bundle 𝔤_{E} ≅ \underline{ℝ} ⊕ ξ admits a connection, reducible with respect to this splitting, given by the trivial connection on \underline{ℝ} and a g-anti-self-dual connection on the line bundle ξ if and only if c_{1}(ξ) · [ω_{g}] = 0, where ω_{g} is a g-self-dual, harmonic two-form.

Recall that any (real) linear representation space V of S^{1} can be decomposed as V = V_{ℂ}⊕V_{ℝ}, where S^{1} acts trivially on the real space V_{ℝ} and by a direct sum of one-dimensional representations on the complex space V_{ℂ}. Furthermore, the only linear irreducible representation which is free away from the origin is the standard representation on ℂ. To determine the possible dimensions for the cohomology of the elliptic deformation complex at a reducible anti-self-dual SO(3) connection, we shall make use of the following general observation of Freed and Uhlenbeck:

Proposition 2.7. [24, Proposition 4.9] Let X be a closed, oriented four-manifold with Riemannian metric g and b^{1}(X) = 0. Let A be a reducible, unitary, projectively antiself-dual connection on a rank-two, Hermitian bundle E over X. If H_{A}^{•} is the cohomology of the deformation complex for the induced reducible, anti-self-dual SO(3) connection on 𝔤_{E}, then there are real isomorphisms

H_{A}^{0} ≅ H^{0}(X; ℝ) ≅ ℝ, H_{A}^{1} ≅ ℂ^{q}, and H_{A}^{2} ≅ ℂ^{p} ⊕ H^{+}(X; ℝ),

where Stab_{A} ≅ S^{1} acts by the standard action on ℂ^{q}, ℂ^{p}, and by the trivial action on H^{0}(X; ℝ), H^{+}(X; ℝ). Here, Stab_{A} ≅ SO(2) denotes the stabilizer in G_{E} of the anti-self-dual SO(3) connection A^{ad} on 𝔤_{E} and, as usual, G_{E} is the group of unitary automorphisms of E with determinant one.

The decomposition given by Freed-Uhlenbeck can described more explicitly by borrowing from [24, pp. 69–70] and the corresponding discussion for reducible PU(2) monopoles in [17]. Recall from [17, §3.4] that the elliptic deformation complex for the g_{0}-anti-self-dual equation at a reducible connection A = A_{1} ⊕ A_{2} on E = L_{1} ⊕ L_{2}, where A_{2} := A_{e} ⊗ A_{1}^{∗} and L_{2} := (det E) ⊗ L_{1}^{−1}, splits into a tangential and a normal deformation complexes; the tangential complex corresponds to the trivial Stab_{A} ≅ S^{1} action. We can thus apply Lemma 3.17 in [17] to partially compute the cohomology groups H_{A}^{•} of the deformation complex for A on 𝔤_{E},

Ω^{0}(𝔤_{E}) ⟶^{d_{A}} Ω^{1}(𝔤_{E}) ⟶^{d_{A}^{+}} Ω^{+}(𝔤_{E}).

The rolled-up full elliptic deformation complex for A has index −2p_{1}(𝔤_{E}) − 3(1 − b^{1} + b^{+}), that is −2ξ^{2} − 6 for b^{1}(X) = 0 and b^{+}(X) = 1, with

ξ = det E ⊗ L_{1}^{−2} and 𝔤_{E} ≅ iℝ ⊕ ξ. (2.5)

When A is reducible, as above, the tangential and normal deformation complexes are

Ω^{0}(iℝ) ⟶^{d} Ω^{1}(iℝ) ⟶^{d^{+}} Ω^{+}(iℝ), (2.6)

Ω^{0}(ξ) ⟶^{d_{A_{ξ}}} Ω^{1}(ξ) ⟶^{d_{A_{ξ}}^{+}} Ω^{+}(ξ), (2.7)

where A_{ξ} = A_{e} ⊗ 2A_{1}^{∗}. Of course, the cohomology of the tangential deformation complex (2.6) is isomorphic to H^{•}(X; ℝ), so H^{0}(X; ℝ) ≅ ℝ while H^{1}(X; ℝ) = 0 and H^{+}(X; ℝ) ≅ ℝ by hypothesis; thus the index of its rolled-up complex is −2. The condition that H^{1}(X; ℝ) = 0 ensures that reducibles do not occur in obstructed positive-dimensional families and so are isolated points of their moduli spaces. The fact that the reducibles are isolated if and only if H^{1}(X; ℝ) = 0 can also be seen from [8, Proposition 2.2.6]; see [8, Proposition 4.2.15] in addition.

Turning to the normal deformation complex (2.7), we recall from [17, Equation (3.45)] that the (real) index of its rolled-up complex is given by

2n := 2n_{ξ} = −2p_{1}(𝔤_{E}) − (e(X) + σ(X)) = −2ξ^{2} − 2(1 − b^{1} + b^{+}). (2.8)

For b^{1}(X) = 0 and b^{+}(X) = 1, this becomes 2n = −2ξ^{2} − 4. Observe that the rolledup normal deformation complex has positive (real) index when ξ^{2} ≤ −3, zero index when ξ^{2} = −2, and index −2 when ξ^{2} = −1. From condition (2.3) we need only ever consider ξ^{2} ≤ −1 and thus n_{ξ} ≥ 0 (when b^{1}(X) = 0 and b^{+}(X) = 1).

Proposition 2.7 implies that for the normal deformation complex (2.7) we must have

H_{A_{ξ}}^{0} = 0, H_{A_{ξ}}^{1} ≅ ℂ^{p+n}, and H_{A_{ξ}}^{2} ≅ ℂ^{p},

where n = −ξ^{2} − 2 is the complex index of the rolled-up normal complex. Hence,

H_{A}^{0} ≅ ℝ, H_{A}^{1} ≅ ℂ^{p+n}, and H_{A}^{2} ≅ ℂ^{p} ⊕ ℝ. (2.9)

The preceding consequence of Proposition 2.7 can be sharpened if the metric g on X is suitably generic, though this refinement is not essential for our purposes.

Theorem 2.8. [24, Theorem 4.19] Continue the hypotheses and notation of Proposition 2.7, with ξ a complex line bundle over X as in (2.5) and n := −ξ^{2} − 2 ≥ −1, the index of the rolled-up normal deformation complex. If, in addition, the metric g on X is ξ-generic, that is it satisfies [ω(g)] ∈ W_{ξ} but is otherwise generic, then the following hold.

• If n ≥ 0, then H_{A_{ξ}}^{1} ≅ ℂ^{n} and H_{A_{ξ}}^{2} ≅ 0. Thus,

H_{A}^{0} ≅ ℝ, H_{A}^{1} ≅ ℂ^{n}, and H_{A}^{2} ≅ ℝ.

• If n = −1, then H_{A_{ξ}}^{1} ≅ 0 and H_{A_{ξ}}^{2} ≅ ℂ. Thus,

H_{A}^{0} ≅ ℝ, H_{A}^{1} ≅ 0, and H_{A}^{2} ≅ ℂ ⊕ ℝ.

Proof. The argument of Freed-Uhlenbeck in [24, pp. 69–73] leading to the proof of [24, Theorem 4.19] makes no use of the facts that b^{+}(X) = 0, that X is simply-connected, or that E is an SU(2) bundle with c_{2}(E) = 1. Observe that the variations of the metric preserve the curvature, F_{A_{ξ}}. The period points of these metrics thus remain in the wall W_{ξ}. Hence, the conclusions follow directly from [24].

2.6. Kuranishi obstructions in the parameterized moduli space. It remains to extend the remarks of the preceding section concerning singularities in the moduli space M_{κ}^{w}(X, g_{0}) to the case of the one-parameter family of moduli spaces M_{κ}^{w}(X, γ).

The usual Kuranishi model for a neighborhood of [A_{0}] ∈ M_{κ}^{w}(X, g_{0}) results from considering the section A ↦ P_{+}(g_{0})F_{A} of the Hilbert bundle A_{E}(X) ×_{G_{E}} L_{ℓ}^{2}(Λ^{+,g_{0}} ⊗ 𝔤_{E}). If g_{t}, t ∈ [−1, 1], is a smooth path of metrics on T^{∗}X, we may choose a smooth path f_{t} ∈ GL(T^{∗}X) such that g_{t} = f_{t}(g_{0}), t ∈ [−1, 1]; the transformations f_{t} are uniquely determined by g_{t} and g_{0} by requiring the f_{t} to be symmetric. Therefore, we have

P_{+}(g_{t}) = f_{t}^{−1} ∘ P_{+}(g_{0}) ∘ f_{t} : Λ^{2} → Λ^{+,g_{t}}

and so instead of considering the zero-locus of the section (A, t) ↦ P_{+}(g_{t})F_{A} of (A_{E}(X) × [−1, 1]) ×_{G_{E}} L_{ℓ}^{2}(Λ^{+,g_{t}} ⊗ 𝔤_{E}), a Hilbert bundle depending on t, we may equivalently consider the zero-locus of the section (A, t) ↦ P_{+}(g_{0})f_{t}(F_{A}) of the fixed Hilbert bundle (A_{E}(X) × [−1, 1]) ×_{G_{E}} L_{ℓ}^{2}(Λ^{+,g_{0}} ⊗ 𝔤_{E}). The construction of the local Kuranishi model for a neighborhood of [A_{0}, g_{0}] ∈ M_{κ}^{w}(X, γ) then follows in the usual way [8] to give:

Lemma 2.9. Let E be a Hermitian two-plane bundle over a closed, oriented, smooth fourmanifold X with b^{+}(X) = 1 and γ : [−1, 1] → Met(X) a generic path of Riemannian metrics on T^{∗}X. Let [A_{0}, g_{0}] be a point in M_{E}(γ) such that γ(0) = g_{0} and A_{0} is a reducible unitary connection on E. Then A_{0} has stabilizer Stab_{A_{0}} ≅ S^{1} in G_{E} and there are

• An open, S^{1}-invariant neighborhood O = O_{A_{0}} of the origin in H_{A_{0}}^{1} and a positive constant ε = ε_{A_{0}} together with a smooth, S^{1}-equivariant embedding γ = γ_{A_{0},g_{0}},

γ : O × (−ε, ε) ⊂ H_{A_{0}}^{1} ⊕ ℝ ↪ A_{E},

with γ(0, 0) = (A_{0}, 0) and (M_{E} ∩ γ(O × (−ε, ε)))/S^{1} an open neighborhood of [A_{0}, g_{0}] in M_{E}(γ); and

• A smooth, S^{1}-equivariant map χ = χ_{A_{0},g_{0}},

χ : O × (−ε, ε) ⊂ H_{A_{0}}^{1} ⊕ ℝ → H_{A_{0}}^{2}

such that γ restricts to a smoothly-stratified diffeomorphism from (χ^{−1}(0) ∩ (O × (−ε, ε)))/S^{1} onto (M_{E} ∩ γ(O × (−ε, ε)))/S^{1}.

The obstruction map can be defined by

χ(a, t) := Π_{A_{0},g_{0}}P_{+}(g_{0})f_{t}(F (A_{0} + a)),

while the existence of the embedding γ follows from the Banach-space implicit function theorem. Here, Π_{A_{0},g_{0}} is the L^{2}-orthogonal projection from L^{2}(Λ^{+,g_{0}} ⊗ 𝔤_{E}) onto the span of the eigenvectors of d_{A_{0}}^{+,g_{0}}d_{A_{0}}^{∗,g_{0}} on L^{2}(Λ^{+,g_{0}} ⊗ 𝔤_{E}) with eigenvalue less than \frac{1}{2}μ, where μ is the first positive eigenvalue of this Laplacian.

The model described in Lemma 2.9 can be simplified: we may remove the factor H^{+}(X; ℝ) from the obstruction space at the cost of restricting the obstruction map χ to the preimage χ_{ℝ}^{−1}(0) ⊂ ℝ ⊕ ℂ^{n}, where χ_{ℝ} is defined by (2.10).

Lemma 2.10. Continue the notation and hypotheses of Lemma 2.9. Assume that b^{1}(X) = 0 and that A_{0} is a reducible SO(3) connection with respect to the splitting 𝔤_{E} ≅ iℝ ⊕ ξ. Then the following hold.

• If n_{ξ} ≥ 1, then there are an open, S^{1}-invariant neighborhood O of the origin in ℂ^{n} and a positive constant ε together with a smooth, S^{1}-equivariant embedding,

γ : O × (−ε, ε) ⊂ ℂ^{n_{ξ}} ⊕ ℝ ↪ A_{E},

with γ(0, 0) = (A_{0}, 0) and M_{E}(γ)∩γ(O/S^{1}×(−ε, ε)), an open neighborhood of [A_{0}, g_{0}] in M_{E}(γ).

The map γ restricts to a smoothly-stratified diffeomorphism from χ_{ℝ}^{−1}(0) ∩ (O/S^{1} × (−ε, ε)) onto M_{E}(γ) ∩ γ(O/S^{1} × (−ε, ε)). Moreover, there is an S^{1}-equivariant diffeomorphism χ_{ℝ}^{−1}(0) ∩ (O × (−ε, ε)) ≅ ℂ^{n}.

• If n_{ξ} ≤ 0, then [A_{0}, g_{0}] is an isolated point of the parameterized moduli space M_{E}(γ), with vanishing obstruction when n_{ξ} = 0 and obstruction space ℂ when n_{ξ} = −1.

Proof. The proof can be extracted from a similar argument used to prove the wall-crossing formula in [6, Proposition 2.12]. By (2.9), Theorem 2.8, and the fact that the metric g_{0} is ξ-generic, H_{A_{0}}^{1} = ℂ^{n_{ξ}} and H_{A_{0}}^{2} = ℝ when n_{ξ} ≥ 0, while H_{A_{0}}^{1} = 0 and H_{A_{0}}^{2} = ℝ ⊕ ℂ when n_{ξ} = −1. With respect to the splitting Λ^{+} ⊗ 𝔤_{E} ≅ iΛ^{+} ⊕ (Λ^{+} ⊗ ξ), we may write χ = (χ_{ℝ}, χ_{ℂ}), where

\begin{aligned}χ_{ℝ}(z, t) & = ⟨χ(z, t), f_{t}(ω_{0})⟩f_{t}(ω_{0}), \\ & = ⟨f_{t}(F (A_{0} + γ(z, t))), f_{t}(ω_{0})⟩f_{t}(ω_{0}), (z, t) ∈ H_{A_{0}}^{1} ⊕ (−ε, ε). \end{aligned} (2.10)

According to Lemma 2.13 we may assume without loss that the path ω_{t} ≡ f_{t}(ω_{0}) crosses the wall W_{ξ} transversely, so (∂χ_{ℝ}/∂t)(0, 0) ≠ 0. Hence, the map π_{ℂ} : χ_{ℝ}^{−1}(0) ≅ ℂ^{n_{ξ}} is an S^{1}-equivariant diffeomorphism.

Remark 2.11. Suppose ξ ∈ H^{2}(X; ℤ) defines a reduction 𝔤_{E_{0}} ≅ iℝ⊕ξ. Because of Lemma 2.10 we see that, provided b^{1}(X) = 0, we need only consider the following cases where the family of reducibles in M_{κ_{0}}^{w}(X, C) consists of either

• A single point [A_{0}, g_{0}] with vanishing obstruction, when n_{ξ} ≥ 0, or

• A single, isolated point [A_{0}, g_{0}] with obstruction H_{A_{0},g_{0}}^{2} ≅ ℂ, when n_{ξ} = −1.

In the first case we would glue up an open neighborhood U_{0} = Ũ_{0}/S^{1} M_{κ_{0}}^{w}(X, C) of the point [A_{0}, g_{0}]; in the second, U_{0} could simply be the isolated point [A_{0}, g_{0}] itself when gluing up. Here, in either case, we can assume that Ũ_{0} ⊂ A_{κ_{0}}^{w}(X) × Met(X) is contained in the slice (A_{0} +Ker d_{A_{0}}^{∗,g_{0}})×C, so Ũ_{0}/G_{E} = U_{0}/S^{1}. We shall make use of this simplification — namely, that we need only glue up anti-self-dual connections rather than extended anti-self-dual connections even when gluing up non-smooth points — when defining the space of gluing data, Gl_{κ}^{w}(X, U_{0}, Σ, λ_{0}), in §3.

2.7. Paths of metrics and the period map. There is a technical point with regard to the definition of the family of background connections over X, in the presence of a path of metrics, which requires some comment. Suppose g_{±1} are two C^{∞} metrics on T X and ω_{±1} are the corresponding self-dual harmonic two-forms on X (we assume b^{+}(X) = 1). The metrics g_{±1} thus define points [ω_{±1}] in H^{+}(X; ℝ) − {0}, lying in the positive cone Ω_{X} of H^{+}(X; ℝ) since if ω = ω_{−1} or ω_{1} then

⟨[ω] ∪ [ω], [X]⟩ = ∫_{X} ω ∧ ω = ∫_{X} ω ∧ ∗ω = ∥ω∥_{L^{2}(X)}^{2} > 0.

In particular, the points [ω_{±1}] lie in chambers C_{±} of Ω_{X}. If C_{+} = C_{−} = C, that is, these points lie in the same chamber, there is a question of whether a path [ω_{t}] ∈ C, t ∈ [−1, 1], joining [ω_{−1}] to [ω_{1}] can be lifted to a path of metrics g_{t} in Met(X) joining g_{−1} to g_{1}. For this purpose, one needs to consider the ‘period map’

π : Met(X) → ℙ(H^{2}(X; ℝ)) := H^{2}(X; ℝ)/ℝ^{∗}, g ↦ π(g) := H^{+}(g),

where H^{±}(g) are the spaces of self-dual and anti-self-dual harmonic two-forms. As remarked in [36, p. 434] it is not known whether two metrics whose period points lie in the same chamber can be connected by a path of metrics whose period points lie in the same chamber. However, following result of Donaldson will suffice for our needs: ^{4}

Proposition 2.12. [5, §6], [8, Proposition 4.3.14] Let X be a closed, oriented four-manifold and let 𝔾(b^{2}, b^{+}) be the real Grassmann manifold of b^{+}(X)-dimensional subspaces of H^{2}(X; ℝ). Let 𝕌 ⊂ 𝔾(b^{2}, b^{+}) be the open subset of planes on which the intersection form Q of X is positive definite. Let π : Met(X) → 𝕌 be the period map g ↦ H^{+}(g). Let ξ ∈ H^{2}(X; ℤ) be a class such that ξ^{2} < 0 and let W_{ξ} ⊂ 𝕌 = {K ∈ 𝕌 : ξ ⊥ K}. Then W_{ξ} is a b^{+}(X)- codimensional submanifold and the period map π is transverse to W_{ξ}.

When b^{+}(X) = 1 then 𝔾(b^{2}, b^{+}) = ℙ(H^{2}(X; ℝ)), 𝕌 = Ω_{X}, and W_{ξ} is a wall. For the purposes of proving the wall-crossing formula, it is enough to consider paths of metrics satisfying the conclusions of the following lemma, which comprise the preamble to Theorem 5.2.2 in [36].

Lemma 2.13. Let X be a closed, oriented, smooth four-manifold with b^{+}(X) = 1 and suppose that C_{±} are chambers in Ω_{X} separated by a single geometric wall W. Then there is a generic path of smooth metrics γ : [−1, 1] → Met(X), t ↦ g_{t}, with the following properties:

1. Fix a class w ∈ H^{2}(X; ℤ) such that no SO(3) bundle V over X with w_{2}(V) = w (mod 2) admits a flat connection. Then for every U(2) bundle E with c_{1}(E) = w and κ ≡ −\frac{1}{4}p_{1}(𝔤_{E}) > 0, the parameterized moduli space M_{κ}^{w}(X, γ) contains no flat connections and is a C^{∞} manifold away from the reducible connections.

2. For all t ≠ 0, the compactification M̄_{κ}^{w}(X, γ) contains no points whose background connection is reducible or flat.

3. The self-dual two-form ω(g_{0}) lies in a single geometric (w, κ)-wall, W, while ω(g_{t}) ∈ C_{−} for t < 0 and ω(g_{t}) ∈ C_{+} for t > 0.

4. The path of self-dual two-forms ω(g_{t}) defined by γ crosses the wall W transversely.

Proof. We may restrict our attention to an open neighborhood U ⊂ ℙ(H^{2}(X; ℝ)) such that U ⊂ C^{−} ∪W ∪C^{+} and so we can express U as a disjoint union U = U ∩C^{−} ⊔U ∩W ⊔U ∩C^{+}. Furthermore, we choose U so that it only meets a single geometric wall W. The wall W is defined by some, not necessarily unique class ξ ∈ H^{2}(X; ℤ). According to Proposition 2.12, the period map π is transverse to W and so π^{−1}(W) is a smooth, codimension-one submanifold of Met(X). Plainly, we may choose a generic path γ : [−1, 1] → Met(X) whose image is contained in the open subset π^{−1}(U) and which is transverse to π^{−1}(W). Consequently, the induced path ω(γ) ≡ π(γ) in H^{2}(X; ℝ)/ℝ^{+} is transverse to W. Finally, we may choose γ short enough that ω(g_{t}) ∈ C_{−} for all t < 0 and ω(g_{t}) ∈ C_{+} for all t > 0.

2.8. Links of reducibles and the computational problem. It remains to state the computational problem which concerns the present article and its sequels. Given the situation of Lemma 2.13 one can prove (see [36, Theorem 5.2.2], though we shall give a different argument elsewhere) the following local wall-crossing result:

4The statement of Proposition 2.12 is taken from [5, p. 336]: the period map defined there is precisely dual to the one considered in [8].

D_{X,g_{+1}}^{w}(z) − D_{X,g_{−1}}^{w}(z) = ∑_{ξ}(−1)^{(w−ξ)^{2}/4}δ_{ξ}^{w}(z), z ∈ 𝔸(X),

where the sum is over all ξ ∈ H^{2}(X; ℤ) which define the geometric (w, κ)-wall W containing ω(g_{0}) and which satisfy ξ · ω(g_{−1}) < 0 < ξ · ω(g_{1}), with deg z = 8κ − 3(2 − b^{1}). The terms δ_{ξ}^{w}(z) have the following properties:

• δ_{ξ}^{w} is independent of where the path γ crosses the wall W,

• δ_{ξ}^{w} changes sign when the direction of crossing is reversed,

• δ_{−ξ}^{w} = (−1)^{ξ^{2}}δ_{ξ}^{w}. It follows easily from these properties that D_{X,g}^{w}(z) = D_{X,g′}^{w}(z) for any pair of metrics g, g′ whose period points lie in the same chamber C of Ω_{X}, so D_{X,C}^{w}(z) = D_{X,g}^{w}(z) is well-defined and the preceding wall-crossing formula can be replaced by its global analogue:

D_{X,C_{+}}^{w}(z) − D_{X,C_{−1}}^{w}(z) = ∑_{ξ}(−1)^{(w−ξ)^{2}/4}δ_{ξ}^{w}(z), z ∈ 𝔸(X),

with C_{±} in the same component of Ω_{X} and the sum is over all ξ ∈ H^{2}(X; ℤ) which define geometric (w, κ)-walls satisfying ξ · C_{−} < 0 < ξ · C_{+}, with deg z = 8κ − 3(2 − b^{1}).

Technicalities aside, each term δ_{ξ}^{w}(z) is defined by pairings

δ_{ξ}^{w}(z) = ⟨μ(z), [L̄_{κ,ξ}^{w}]⟩,

where μ : H_{i}(X; ℚ) → H^{4−i}(B_{κ}^{w}(X) × γ; ℚ) and L_{κ,ξ}^{w} is the “link” in M_{κ}^{w,∗}(X, γ) of the level containing the reducible SO(3) connection defined by w, ξ, κ,

[A_{0}] × Sym^{ℓ}(X) ⊂ M̄_{κ}^{w}(X, γ),

while L̄_{κ,ξ}^{w} is its closure in M̄_{κ}^{w}(X, γ). Here, γ is a path satisfying the conclusions of Lemma 2.13 such that A_{0} is a reducible g_{0}-anti-self-dual SO(3) connection on an SO(3) bundle 𝔤_{E_{0}} with w_{2}(𝔤_{E_{0}}) = w (mod 2) and −\frac{1}{4}p_{1}(𝔤_{E_{0}}) = κ − ℓ. The reduction of the SO(3) bundle 𝔤_{E_{0}} is defined by the class ξ: we have E_{0} = L ⊕ (w ⊗ L^{−1}) and 𝔤_{E_{0}} ≅ ξ ⊕ \underline{ℝ}, where ξ = w ⊗ L^{−2}, according to our standard convention, and thus p_{1}(𝔤_{E_{0}}) = ξ^{2}.

The heart of the problem then is to show, using a more precise version of this “definition”, that the δ_{ξ}^{w} depend at most on the cohomology classes w, ξ ∈ H^{2}(X; ℤ), the homotopy type of X, and the degree of z. The above computation has been carried out directly when ℓ = 0, 1, 2 (see [6], [40], [35], [36], [64]), which verifies the conjecture in these cases, so our task is to extend this work so we may conclude that the δ_{ξ}^{w}(z) have the desired invariance properties even if an explicit formula cannot be deduced by purely topological methods.

3. Splicing connections

We construct a splicing map γ′ which will give a homeomorphism from the total space of a gluing data bundle into an open neighborhood in B̄_{κ}^{w}(X) of a level M_{κ−ℓ}^{w}(X) × Σ, where Σ ⊂ Sym^{ℓ}(X) is a smooth stratum. In this section we keep the Riemannian metric g on T X fixed, though we shall later allow the metrics g to vary in a precompact open subset of the space Met(X) of all C^{∞} Riemannian metrics on T X. When defining our spaces of connections and gauge transformations, we shall always choose to work L_{k}^{2} connections modulo L_{k+1}^{2} gauge transformations with k ≥ 4: though only k ≥ 2 is strictly necessary, we shall occasionally need to consider estimates whose constants depend on the C^{0} norm of F_{A}^{+}. Thus, for the remainder of this article and its sequels [14], [15], [16], we shall assume k ≥ 4 in order to make use of the Sobolev embedding L_{4}^{2} ⊂ C^{0}.

3.1. Uhlenbeck neighborhoods. In order to associate sequences of mass centers and scales to a sequence of points [A_{α}] in M_{E}(X, g) converging to a point [A_{0}, x] in M̄_{E}(X, g), we need a suitable finite open cover of M̄_{E}(X, g); for simplicity, we take C = {g}. Let ϱ be the injectivity radius of the Riemannian manifold (X, g) and let dist_{g}(·, ·) be the distance function on X defined by the metric g.

Definition 3.1. Suppose [A_{0}, x] is a point in M̄_{E}(X, g), where A_{0} is a connection on a Hermitian, rank-two bundle E_{0} over X with c_{2}(E_{0}) < c_{2}(E), c_{1}(E_{0}) = c_{1}(E) and ε, δ, M are positive constants with M ≥ 1, and ε < 2π, and Mδ ≤ \frac{1}{4}\{1, ϱ, d_{x}\}, where d_{x} := min_{i≠j} dist(x_{i}, x_{j}) is the minimum distance separating the distinct points x_{i} of the multiset x = (x_{1}, …, x_{m}). Assume Mδ is small enough that

∫_{B(x_{i},Mδ)} |F_{A_{0}}|^{2} dV < \frac{1}{4}ε^{2}.

Then [A] ∈ M_{E}(X, g) lies in an (ε, δ, M) Uhlenbeck open neighborhood U of [A_{0}, x] if the following hold:

• There is an L_{k+1,loc}^{2} bundle map u : E|_{X\{x}} → E_{0}|_{X\{x}} such that the connections u(A) and A_{0} are L_{k}^{2} close on X \ B_{x}(\frac{1}{2}δ):

∥u(A) − A_{0}∥_{L_{k}^{2}(A_{0};X\B(x,δ/2))} < ε;

• The measure |F_{A}|^{2} is close to |F_{A_{0}}|^{2} + 8π^{2}κ_{x}δ_{x}:

|∫_{B(x_{i},δ)} (|F_{A}|^{2} − |F_{A_{0}}|^{2} − 8π^{2}κ_{i}δ_{x_{i}}) dV | < \frac{1}{4}ε^{2},

|∫_{Ω(x_{i};δ,Mδ)} (|F_{A}|^{2} − |F_{A_{0}}|^{2}) dV | < \frac{1}{4}ε^{2},

for i = 1, …, m and where (x_{1}, …, x_{m}) is a representative for x with multiplicities κ_{1}, …, κ_{m}.

The constraints on the curvature F_{A} in Definition 3.1 imply that

|8π^{2}κ_{i} − ∫_{B(x_{i},δ)} |F_{A}|^{2} dV | < \frac{1}{2}ε^{2},

(3.1)  ∫_{Ω(x_{i};δ,Mδ)} |F_{A}|^{2} dV < \frac{1}{2}ε^{2},

|8π^{2}κ_{i} − ∫_{B(x_{i},Mδ)} |F_{A}|^{2} dV | < ε^{2}.

Observe that no cutoff functions are required to smooth out the implied characteristic functions of the balls with center x_{i} and radii δ, Mδ above as Definition 3.1 rules out the possibility of the curvature F_{A} concentrating on their boundaries.

3.2. Gluing data. In this section we describe the gluing-data bundles we shall need to parameterize an open neighborhood in M̄_{κ}^{w}(X, C) of a level (M_{κ−ℓ}^{w}(X, C) × Σ) ∩ M̄_{κ}^{w}(X, C).

3.2.1. Connections over the four-sphere. We begin by defining the required gluing data associated with the four-spheres: this will, essentially, define the fibers of our gluing-data bundles. Let κ ≥ 1 be an integer and let S^{4} have its standard round metric of radius one. Let E be a Hermitian, rank-two vector bundle over S^{4} with c_{2}(E) = κ and let 𝔤_{E} = 𝔰𝔲(E) be the corresponding SO(3) bundle with −\frac{1}{4}p_{1}(𝔤_{E}) = κ.

A choice of frame f in the principal SO(4) frame bundle Fr(T S^{4}) for T S^{4}, over the north pole n ∈ S^{4}, defines a conformal diffeomorphism,

φ_{n} : ℝ^{4} → S^{4} \ {s}, (3.2)

that is inverse to a stereographic projection from the south pole s ∈ S^{4} ⊂ ℝ^{5}; let y( ·) : S^{4} \ {s} → ℝ^{4} be the corresponding coordinate chart.

Definition 3.2. [55, pp. 343–344] Continue the notation of the preceding paragraph. The center z = z([A], f) ∈ ℝ^{4} and the scale λ = λ([A], f) of a point [A] ∈ B_{E}(S^{4}) = B_{κ}(S^{4}) are defined by

(3.3)  z([A], f) := \frac{1}{8π^{2}κ} ∫_{ℝ^{4}} y|F_{A}|^{2} d^{4}y,
(3.4)  λ([A], f)^{2} := \frac{1}{8π^{2}κ} ∫_{ℝ^{4}} |y − z([A], f)|^{2}|F_{A}|^{2} d^{4}y.

A point [A] is centered if z([A], f) = 0 and λ([A], f) = 1. An ideal point [A, x] ∈ B_{κ−ℓ}(S^{4})× Sym^{ℓ}(S^{4}) is centered if z([A, x], f) = 0 and λ([A, x], f) = 1, where the center and scale are defined by replacing |F_{A}|^{2} with |F_{A}|^{2} + 8π^{2} ∑_{x∈x} δ_{x} in equations (3.3) and (3.4).

For any (λ, z) ∈ ℝ^{+} × ℝ^{4}, we define a conformal diffeomorphism of ℝ^{4} by

f_{λ,z} : ℝ^{4} → ℝ^{4}, y ↦ (y − z)/λ. (3.5)

It is easy to see that the point [(f_{λ,z}^{−1})^{∗}A] is centered if z = z([A], f) and λ = λ([A], f). Via the standard action of SO(4) ⊂ GL(ℝ^{4}) on ℝ^{4} and the conformal diffeomorphism φ_{n}^{−1} : S^{4} \ {s} → ℝ^{4}, we obtain a homomorphism from SO(4) into the group of conformal diffeomorphisms of S^{4} fixing the south pole s. The group SO(4) acts on the quotient space of connections based at the south pole, B_{κ}^{s}(S^{4}) ≅ A_{κ}(S^{4}) ×_{G_{E}} Fr(𝔤_{E})|_{s}, by

SO(4) × B_{κ}^{s}(S^{4}) → B_{κ}^{s}(S^{4}), (u, [A, q]) ↦ u · [A, q] = [(u^{−1})^{∗}A, q]. (3.6)

The assignment f ↦ z([A], f) is SO(4)-equivariant, that is, z([A], uf) = uz([A], f) for u ∈ SO(4), and so λ[A] = λ([A], f) is independent of the choice of f ∈ Fr(T S^{4})|_{n}. The center and scale functions define a smooth map (z, λ) : B_{κ}(S^{4}) → ℝ^{4} ×ℝ^{+} and the preimage of (0, 1) ∈ ℝ^{4} × ℝ^{+}, namely the codimension-five submanifold of centered connections

B_{κ}^{⋄}(S^{4}) := {[A] : z[A] = 0 and λ[A] = 1} ⊂ B_{κ}(S^{4}), (3.7) serves as a canonical slice for the action of ℝ^{4} × ℝ^{+} on B_{κ}(S^{4}); we simply write z[A] when the frame f for T S^{4}|_{n} is understood and employ the decoration ⋄ to indicate a subspace of centered connections. Since the action of SO(4) preserves the north and south poles of S^{4}, it acts on B_{κ}^{⋄,s}(S^{4}), and so B_{κ}^{⋄,s}(S^{4}) admits an action of SO(3) × SO(4), with SO(3) varying the frame of 𝔤_{E}|_{s}.

A simple a priori decay estimate for the curvature of connections is provided by the Chebychev inequality:

Lemma 3.3. Let [A] ∈ B_{κ}(S^{4}) and suppose z([A], f) = 0. If λ = λ[A] and R > 0, then

\frac{1}{8π^{2}κ} ∫_{|y|≥Rλ} |F_{A}|^{2} d^{4}y ≤ R^{−2}.

Proof. From equation (3.4) we see that

λ^{2} ≥ \frac{1}{8π^{2}κ} ∫_{|y|≥Rλ} |y|^{2}|F_{A}|^{2} d^{4}y ≥ \frac{R^{2}λ^{2}}{8π^{2}κ} ∫_{|y|≥Rλ} |F_{A}|^{2} d^{4}y

and this gives the required bound.

An immediate consequence of the Chebychev inequality is that if A is a centered antiself-dual connection over S^{4}, with second Chern number κ so its energy is equal to 8π^{2}κ, then A cannot bubble outside the ball B(0, 1/2π) ⊂ ℝ^{4}, as it has most energy 4π^{2} in this region by Lemma 3.3.

We let M_{κ}^{⋄}(S^{4}) ⊂ B_{κ}^{⋄}(S^{4}) denote the subspace of anti-self-dual connections, centered at the north pole, while M_{κ}^{s}(S^{4}) ⊂ B_{κ}^{s}(S^{4}) is the subspace of anti-self-dual connections, framed at the south pole. Finally, we let M_{κ}^{⋄,s}(S^{4}) ⊂ B_{κ}^{⋄,s}(S^{4}) denote the subspace of anti-self-dual connections, which framed at the south pole and centered at the north pole. The definition of Uhlenbeck convergence (Definition 2.3) for sequences of points [A_{α}] in B_{κ}(S^{4}) generalizes easily to the case of sequences [A_{α}, q_{α}] in B_{κ}^{s}(S^{4}), when Fr(𝔤_{E}|_{s}) ≅ SO(3) is given its usual topology. Since Lemma 3.3 implies that points in M⋄, s_{κ}(S^{4}) cannot bubble near the south pole, the proof of [8, Theorem 4.4.3] adapts with no change to show that M_{κ}^{⋄,s}(S^{4}) has compact closure M̄_{κ}^{⋄,s}(S^{4}) in the space of framed, ideal anti-self-dual connections,

⋃_{ℓ=0}^{κ} (M_{κ−ℓ}^{s}(S^{4}) × Sym^{ℓ}(S^{4})) .

Note that it is the ideal points [A_{0}, x] in M̄_{⋄,s}^{s}κ(S^{4}) which are centered in the sense of Definition 3.2 and not the background connections A_{0}. Note also that the points in S^{4} = ℝ^{4} ∪ {∞} representing the multisets x will be constrained to lie in the ball B̄(0, 1/2π) by Lemma 3.3.

Observe that since the bundle map u in the definition of an Uhlenbeck neighborhood respects the SO(3) action on frames, there is a global action of SO(3) on the space M̄_{κ}^{⋄,s}(S^{4}). This action is free on all strata except [Θ] × Sym^{κ}(S^{4}), where the background connection is trivial. In addition, the SO(4) action on M_{κ}^{⋄,s}(S^{4}) extends over M̄_{κ}^{⋄,s}(S^{4}).

3.2.2. Gluing data bundles. We now define bundles of gluing data, Gl_{κ}^{w}(X, U_{0}, Σ, λ_{0}), associated to a choice of bundle data (κ, w), smooth stratum Σ ⊂ Sym^{ℓ}(X), subset C ⊂ GL(T^{∗}X), open neighborhood U_{0} ⊂ M_{κ_{0}}^{w}(X, C), metric g on T X, and positive constant λ_{0}. Here, C ⊂ GL(T^{∗}X) is a compact, connected, smooth submanifold-with-boundary containing the identity: the pair (C, g) defines a compact, connected, smooth submanifold-with-boundary of Met(X) which we also denote, when no confusion can arise, by C. Our definition is motivated by those of Friedman-Morgan [25, §§3.4.2–3.4.4], Kotschick-Morgan [36, §4], and Taubes [55], [56].

Choose κ ∈ \frac{1}{4}ℤ and w ∈ H^{2}(X; ℤ) according to condition (2.1) and let E be the corresponding Hermitian, rank-two vector bundle over X defined by condition (2.4). Let ℓ be an integer such that 1 ≤ ℓ ≤ ⌊κ⌋ and let Σ ⊂ Sym^{ℓ}(X) be a smooth stratum, defined by an integer partition κ_{1} ≥ ⋯ ≥ κ_{m} ≥ 1 of ℓ such that κ_{1} + ⋯ + κ_{m} = ℓ. Let E_{0} be the Hermitian, rank-two vector bundle over X with det E_{0} = det E and c_{2}(E_{0}) = c_{2}(E) − ℓ. Let Fr(T X) denote the principal SO(4) bundle of oriented, g-orthonormal frames for T X. Suppose Ũ_{0} ⊂ A_{E_{0}}(X) × C projects to an open subset U_{0} ⊂ M_{E_{0}}(X, C). Observe that the fibered product

Ũ_{0} ×_{G_{E_{0}}} ∏_{i=1}^{m} (Fr(𝔤_{E_{0}}) ×_{X} Fr(T X)) → U_{0} × ∏_{i=1}^{m} X (3.8)

is a principal ∏_{i=1}^{m}(SO(3) × SO(4)) bundle. Let Δ ⊂ ∏_{i=1}^{m} X be the full diagonal, that is, the subset of m-tuples of points in X where some point occurs with multiplicity at least two. Let 𝔖 = 𝔖_{Σ} = 𝔖(κ_{1}, …, κ_{m}) be the largest subgroup of the symmetric group on m symbols which preserves the multiplicities. The group 𝔖(κ_{1}, …, κ_{m}) acts freely on ∏_{i=1}^{m} X − Δ and we have an identification

(∏_{i=1}^{m} X − Δ) /𝔖(κ_{1}, …, κ_{m}) = Σ ⊂ Sym^{ℓ}(X).

The group 𝔖(κ_{1}, …, κ_{m}) acts on the fibered product bundle (3.8) to give a principal bundle

Fr_{E_{0}}(X, U_{0}, Σ) → U_{0} × Σ, (3.9)

with total space Fr_{κ}^{w}(X, U_{0}, Σ) = Fr_{E_{0}}(X, U_{0}, Σ), where

(3.10) Fr_{E_{0}}(X, U_{0}, Σ) := (Ũ_{0} ×_{G_{E_{0}}} ∏_{i=1}^{m} (Fr(𝔤_{E_{0}}) ×_{X} Fr(T X))) |_{∏_{i=1}^{m} X−Δ}

and structure group

G(Σ) = G(κ_{1}, …, κ_{m}) := (∏_{i=1}^{m}(SO(3) × SO(4))) ⋊ 𝔖(κ_{1}, …, κ_{m}). (3.11)

Let S^{4} have its standard round metric of radius one, let E_{1}, …, E_{m} be Hermitian, rank-two vector bundles over S^{4} with c_{2}(E_{j}) = κ_{j} for j = 1, …, m, and let 𝔤_{E_{j}} ≡ 𝔰𝔲(E_{j}) be the corresponding SO(3) bundles with −\frac{1}{4}p_{1}(𝔤_{E_{j}}) = κ_{j}. Denote ℝ^{+} := (0, ∞). The smooth manifold

Z(Σ) = Z(κ_{1}, …, κ_{m}) := ∏_{i=1}^{m} (M̃_{E_{i}}^{⋄} ×_{G_{E_{i}}} Fr(𝔤_{E_{i}})|_{s} × ℝ^{+}) (3.12)

carries a natural action of G(κ_{1}, …, κ_{m}) coming from the action of SO(4) rotating S^{4} and the action of SO(3) on the choice of frame for 𝔤_{E_{i}}|_{s} in each factor, and the action of 𝔖(κ_{1}, …, κ_{m}) permuting the factors whose underlying bundles are isomorphic. The bundle Fr(T X) has its usual right SO(4) action given by

SO(4) × Fr(T X) → Fr(T X), (u, f) ↦ fu^{−1}. (3.13)

We form the fibered product Gl_{κ}^{w}(X, U_{0}, Σ) = Gl_{E}(X, U_{0}, Σ), where

(3.14) Gl_{E}(X, U_{0}, Σ) := Fr_{E_{0}}(X, U_{0}, Σ) ×_{G(Σ)} Z(Σ),

together with the natural projection map

π : Gl_{E}(X, U_{0}, Σ) → U_{0} × Σ. (3.15)

This is a locally trivial fiber bundle with structure group G(κ_{1}, …, κ_{m}). We let Z(Σ, λ_{0}) ⊂ Z(Σ) denote the open subspace obtained by replacing ℝ^{+} by (0, λ_{0}) in the definition (3.12) and let Gl_{κ}^{w}(X, U_{0}, Σ, λ_{0}) = Gl_{E}(X, U_{0}, Σ, λ_{0}) denote the corresponding gluing data bundle.

3.2.3. Compactification of the gluing data. The gluing-data bundle Gl_{E_{0}}(X, U_{0}, Σ) has two sources of non-compactness, the non-compact fiber Z(Σ) and the non-compact base, Σ. We now describe a compactification of Gl_{E_{0}}(X, U_{0}, Σ). It is a trivial matter to replace the precompact neighborhood U_{0} ⊂ M_{E_{0}}(X, C) by its compactification Ū_{0}, so we assume that U_{0} is itself compact.

We first describe the closure of the fibers. The definition of Z(Σ) uses the space M_{E_{i}}^{⋄,s}(S^{4}) and an dilation parameter lying in ℝ^{+} = (0, ∞). The dilation action (3.5) maps points in M_{E_{i}}^{⋄,s} × (0, ∞) to framed connections on 𝔤_{E_{i}}. This action extends naturally to M̄_{E_{i}}^{⋄,s}(S^{4}) × (0, ∞). The Uhlenbeck limit of these connections as the dilation parameter goes to zero is given by [Θ, x] where Θ is a trivial connection and x ∈ Sym^{κ}(S^{4}) is represented by (n, n, …, n), where n is the north pole. Thus, we define a ccompletion of

M̄_{E_{i}}^{⋄,s}(S^{4}) × (0, ∞),

by the cone c(M̄_{E_{i}}^{⋄,s}(S^{4})), where the cone point corresponds to zero dilation parameter. Then, let

Z̄(Σ) = Z̄(κ_{1}, …, κ_{m}) := ∏_{i=1}^{m} c(M̄_{E_{i}}^{⋄,s}(S^{4})).

The fibers of the gluing data bundle are then compactified if we work with the space:

(3.16)  Ḡl_{E}(X, U_{0}, Σ) := Fr_{E_{0}}(X, U_{0}, Σ) ×_{G(Σ)} Z̄(Σ).

As the only difference between this space and Gl_{E}(X, U_{0}, Σ) is given by the change in the fiber from Z(Σ) to Z̄(Σ), the projection (3.15) extends to

π : Ḡl_{E}(X, U_{0}, Σ) → U_{0} × Σ,

and this extension is still a locally trivial fiber bundle with the same structure group. We again let Ḡl_{E}(X, U_{0}, Σ, λ_{0}) be the open subspace given by requiring all dilation parameters to be less than λ_{0}.

We deal with the non-compact base, Σ, by replacing Fr_{E_{0}}(X, U_{0}, Σ) with

Fr_{E_{0}}(X, U_{0}, Σ̄) := Ũ_{0} ×_{G_{E_{0}}} ∏_{i=1}^{m} (Fr(𝔤_{E_{0}}) ×_{X} Fr(T X)) . (3.17)

The quotient X^{m}/𝔖_{Σ} can be identified with Σ̄, the closure of Σ in Sym^{ℓ}(X), and so there is a projection

Fr_{E_{0}}(X, U_{0}, Σ̄) → U_{0} × Σ̄.

However, because the action of 𝔖_{Σ} is not free over Σ̄\Σ, this extension is not a fiber bundle.

The splicing and gluing maps are defined on a subspace of the following extension of the gluing data bundle,

(3.18)  Ḡl_{E}(X, U_{0}, Σ̄) := Fr_{E_{0}}(X, U_{0}, Σ̄) ×_{G(Σ)} Z̄(Σ),

as we shall see in §3.3.3.

3.3. Splicing connections over X. In this section we describe the splicing construction for SO(3) connections on 𝔤_{E} via a partition of unity on X and define the splicing maps γ′ for connections. Our description is largely motivated by those of Taubes in [52], [53], [54], [55, §4], [56]. We keep the notation of §3.2.

3.3.1. Cutting and splicing connections. Fix a smooth stratum Σ ⊂ Sym^{ℓ}(X), let x ∈ Σ, and choose r_{0} = r_{0}(x) > 0 to be one half the smaller of

• The minimum geodesic distance between distinct points x_{i}, x_{j} of the representative (x_{1}, …, x_{m}) of x, and

• The injectivity radius of the Riemannian manifold (X, g). A choice of L_{k}^{2} connection A_{0} on 𝔤_{E_{0}} and SO(3) frames p_{i} ∈ Fr(𝔤_{E_{0}})|_{x_{i}} define L_{k+1}^{2} local sections of Fr(𝔤_{E_{0}}),

σ_{i} = σ_{i}(A_{0}, p_{i}) : B(x_{i}, r_{0}) → Fr(𝔤_{E_{0}}), i = 1, …, m,

using parallel transport along radial geodesics emanating from x_{i} together with smoothing for sections of 𝔤_{E_{0}} using the heat kernel exp(−td_{A_{0}}^{∗}d_{A_{0}}) for small t, as described in [21, §A.1]; see also the remarks in [56, p. 177]. Let

𝔤_{E_{0}}|_{B(x_{i},r_{0})} ≃ B(x_{i}, r_{0}) × SO(3) (3.19)

be the L_{k+1}^{2} local trivializations defined by the sections σ_{i}.

Define a smooth cutoff function χ_{x_{0},ε} : X → [0, 1] by setting

χ_{x_{0},ε}(x) := χ(dist(x, x_{0})/ε) (3.20)

where χ : ℝ → [0, 1] is a smooth function such that χ(t) = 1 for t ≥ 1 and χ(t) = 0 for t ≤ 1/2. Thus, we have

χ_{x_{0},ε}(x) = \begin{cases}1 & for x ∈ X − B(x_{0}, ε), \\ 0 & for x ∈ B(x_{0}, ε/2). \end{cases}

We define a cut-off connection on the bundle 𝔤_{E_{0}} over X by setting

(3.21) χ_{x,4\sqrt{λ}}A_{0} := \begin{cases}A_{0} & over X − ∪_{i=1}^{m}B(x_{i}, 4\sqrt{λ_{i}}), \\ Γ + χ_{x_{i},4\sqrt{λ_{i}}}σ_{i}^{∗}A_{0} & over Ω(x_{i}, 2\sqrt{λ_{i}}, 4\sqrt{λ_{i}}), \\ Γ & over B(x_{i}, 2\sqrt{λ_{i}}), \end{cases}

where Γ denotes the product connection on B(x_{i}, r_{0}) × SO(3), while x = (x_{1}, …, x_{m}) and λ = (λ_{1}, …, λ_{m}).

Let A_{i} be centered L_{k}^{2} connections on the bundles 𝔤_{E_{i}} over S^{4}. Let

τ_{i} = τ_{i}(A_{i}, q_{i}) : S^{4} \ {n} → Fr(𝔤_{E_{i}}), i = 1, …, m,

be the analogously-defined L_{k+1}^{2} local sections induced by the connections A_{i} ∈ A_{E_{i}} and SO(3) frames q_{i} ∈ Fr(𝔤_{E_{i}})|_{s}. Let

𝔤_{E_{i}}|_{S^{4}\{n}} ≃ S^{4} \ {n} × SO(3) (3.22)

be the L_{k+1}^{2} local trivializations defined by the τ_{i}. Let δ_{λ} : S^{4} → S^{4} be the conformal diffeomorphism of S^{4} defined by

δ_{λ} := φ_{n} ∘ λ ∘ φ_{n}^{−1}, (3.23)

where λ : ℝ^{4} → ℝ^{4} is the dilation given by y ↦ y/λ, so δ_{λ} is a conformal diffeomorphism of S^{4} which fixes the north and south poles. We define cut-off, rescaled connections on the bundles 𝔤_{E_{i}} over S^{4}, with mass centers at the north pole, by setting

(3.24) (1 − χ_{x_{i},\sqrt{λ_{i}}/2})δ_{λ_{i}}^{∗}A_{i} := \begin{cases}δ_{λ_{i}}^{∗}A_{i} & over φ_{n}(B(0, \frac{1}{4} \sqrt{λ_{i}})), \\ Γ + (1 − χ_{x_{i},\sqrt{λ_{i}}/2})τ_{i}^{∗}δ_{λ_{i}}^{∗}A_{i} & over φ_{n}(Ω(0, \frac{1}{4} \sqrt{λ_{i}}, \frac{1}{2} \sqrt{λ_{i}})), \\ Γ & over S^{4} − φ_{n}(B(0, \frac{1}{2} \sqrt{λ_{i}})), \end{cases}

where Γ denotes the product connection on (S^{4}−{n})×SO(3). Note that τ_{i}^{∗}δ_{λ_{i}}^{∗}A_{i} = δ_{λ_{i}}^{∗}τ_{i}^{∗}A_{i}, since the section τ_{i} is defined by parallel translation from the south pole via the connection A_{i}.

A choice of SO(4) frames f_{i} ∈ Fr(T X)|_{x_{i}} defines coordinate charts

φ_{i}^{−1} : B(x_{i}, r_{0}) ⊂ X → ℝ^{4}, i = 1, …, m,

via the exponential maps exp_{f_{i}} : B(0, r_{0}) ⊂ T X|_{x_{i}} → X. The orientation-preserving diffeomorphism φ_{i} ∘ φ_{n}^{−1} identifies the annulus φ_{n}(Ω(0, \frac{1}{2} \sqrt{λ_{i}}, 2\sqrt{λ_{i}})) in S^{4},

Ω (0, \frac{1}{2} \sqrt{λ_{i}}, 2\sqrt{λ_{i}}) := φ_{n} (\{x ∈ ℝ^{4} : \frac{1}{2} \sqrt{λ_{i}} < |x| < 2\sqrt{λ_{i}}\}) ⊂ ℝ^{4}

with the annulus in X,

Ω (x_{i}, \frac{1}{2} \sqrt{λ_{i}}, 2\sqrt{λ_{i}}) := \{x ∈ X : \frac{1}{2} \sqrt{λ_{i}} < dist_{g}(x, x_{i}) < 2\sqrt{λ_{i}}\} ⊂ X.

We define a glued-up SO(3) bundle 𝔤_{E} over X by setting

(3.25) 𝔤_{E} = \begin{cases}𝔤_{E_{0}} & over X − ∪_{i=1}^{m}B(x_{i}, \frac{1}{2} \sqrt{λ_{i}}), \\ 𝔤_{E_{i}} & over B(x_{i}, 2\sqrt{λ_{i}}). \end{cases}

The bundles 𝔤_{E_{0}} and 𝔤_{E_{i}} are identified over the annuli Ω(x_{i}, \frac{1}{2} \sqrt{λ_{i}}, 2\sqrt{λ_{i}}\}) in X via the isomorphisms of SO(3) bundles defined by the orientation-preserving diffeomorphisms φ_{i} ∘ φ_{n}^{−1}, identifying the annuli Ω(x_{i}, \frac{1}{2} \sqrt{λ_{i}}, 2\sqrt{λ_{i}}\}) with the corresponding annuli in S^{4} and the SO(3) bundle maps defined by the trivializations (3.19) and (3.22). Define a spliced L_{k}^{2} connection A on 𝔤_{E} by setting

(3.26) A := \begin{cases}A_{0} & over X − ∪_{i=1}^{m}B(x_{i}, 4\sqrt{λ_{i}}), \\ Γ + χ_{x_{i},4\sqrt{λ_{i}}}A_{0} + (1 − χ_{x_{i},\sqrt{λ_{i}}/2})δ_{λ_{i}}^{∗}A_{i} & over Ω(x_{i}, \frac{1}{4} \sqrt{λ_{i}}, 4\sqrt{λ_{i}}), \\ δ_{λ_{i}}^{∗}A_{i} & over B(x_{i}, \frac{1}{4} \sqrt{λ_{i}}), i = 1, …, m, \end{cases}

where the cut-off connections are defined as above; the bundle and annulus identifications are understood. Also implicit in definition (3.26) are the following constraints on the scales, given an m-tuple of distinct points, (x_{1}, …, x_{m}),

8\sqrt{λ_{i}} + 8\sqrt{λ_{j}} < dist_{g}(x_{i}, x_{j}), i ≠ j, (3.27)

representing a multiset x ∈ Σ. The pre-splicing map, γ′′, is defined on the open subset of the gluing-data bundle, Gl_{κ}^{w}(X, U_{0}, Σ) in definition (3.14), defined by

(3.28) Gl_{κ}^{w,+}(X, U_{0}, Σ) := {π^{−1}([A_{0}], x) ∈ Gl_{κ}^{w}(X, U_{0}, Σ) : (λ, x) obeys (3.27)}.

If we replace Σ in Gl_{κ}^{w}(X, U_{0}, Σ) by a compact subset K ⋐ Σ, so that there is a uniform positive lower bound for dist_{g}(x_{i}, x_{j}), i ≠ j, for all x ∈ K, then we can plainly choose a positive constant λ_{0} = λ_{0}(K, g) such that

Gl_{κ}^{w}(X, U_{0}, K, λ_{0}) ⊂ Gl_{κ}^{w,+}(X, U_{0}, Σ),

as the m-tuples of scales λ = (λ_{1}, …, λ_{m}) present in Gl_{κ}^{w}(X, U_{0}, K, λ_{0}) satisfy the constraints (3.27).

3.3.2. Splicing maps. It remains to summarize what the splicing construction has achieved. The construction, thus far, yields a pre-splicing map giving a smooth embedding

γ′′ : Gl_{κ}^{w,+}(X, U_{0}, Σ) → B_{κ}^{w,∗}(X) × Met(X) (3.29)

The fact that the map is a smooth embedding is a fairly straightforward consequence of the definitions; it will follow from the stronger result in [14] that the gluing map γ gives a smooth embedding of the gluing data. Similarly, the fact that the image of γ′′ contains only irreducible connections is also a fairly easy consequence of the definitions; it is proved in [14].

The simple pre-splicing map we have described in §3.3.1 is not the the map we shall ultimately use to produce almost anti-self-dual connections, however, for the reasons we mentioned in the Introduction. Specifically, we have assumed that the connections on the different copies of S^{4} are anti-self-dual with respect to the standard round metric g_{0} of radius one. In practice, we shall need to first use Theorem 1.1 to produce diffeomorphisms from the spaces M_{κ_{i}}^{s}(S^{4}, g_{0}) to M_{κ_{i}}^{s}(S^{4}, g̃_{i}), where g̃_{i} is an almost round metric on S^{4} which is conformally equivalent to g near the points x_{i} ∈ X. More precisely, we first define families of almost flat metrics g_{i} on ℝ^{4} = (T X)_{x_{i}}, depending on the metric g on X, gluing site x_{i}, and scale λ_{i}, by setting

g_{i} := \begin{cases}χ_{8\sqrt{λ_{i}}}δ + (1 − χ_{8\sqrt{λ_{i}}}) exp_{f_{i}}^{∗} g & on B(0, 8\sqrt{λ_{i}}), \\ & on (T X)_{x_{i}} − B(0, 8\sqrt{λ_{i}}), \end{cases} (3.30)

where χ_{8\sqrt{λ_{i}}} is given by definition (3.20). Here, δ is simply the Euclidean metric on (T X)_{x_{i}} defined by the metric g on T X, noting that (exp_{f_{i}}^{∗} g)(0) = δ. We then obtain a family of conformally equivalent metrics g̃_{i} on S^{4} = (T X)_{x_{i}} ∪ {∞} which is close to the standard metric g_{0} by setting

φ_{n}^{∗}g̃_{i}(x) := \frac{4}{(1 + |x|)^{2}} g_{i}(λ_{i}x), x ∈ ℝ^{4}, (3.31)

where we recall that φ_{n} : ℝ^{4} → S^{4}−{s} is inverse to stereographic projection. The definition of geodesic, normal (or ‘Gaussian’) coordinates implies that on a small ball around the points x_{i} ∈ X, the metric g_{i} is C^{1}-close to g and that the difference between g and g_{i} is C^{2}-bounded, uniformly with respect to the ball radius [3].

Lemma 3.4. Continue the above notation. Then g̃_{i} converges to g_{0} in C^{∞}(S^{4} \ s) as λ_{i} converges to zero.

Hence, the true splicing map,

γ′ : Gl_{κ}^{w,+}(X, U_{0}, Σ) → B_{κ}^{w,∗}(X) × Met(X) (3.32)

will be a composition of the diffeomorphisms,

M_{κ_{i}}^{s}(S^{4}, g_{0}) ≅ M_{κ_{i}}^{s}(S^{4}, g̃_{i}),

followed by the pre-splicing map γ′′ in definition (3.29) via the cut-and-paste operation described in §3.3.1.

3.3.3. Extension of the splicing and gluing maps. It remains to define an extension of the pre-splicing map γ′′, splicing map γ′, and gluing map γ from Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) to Ḡl_{κ}^{w,+}(X, U_{0}, Σ̄, λ_{0}). Thus, given a point p in Ḡl_{κ}^{w,+}(X, U_{0}, Σ̄, λ_{0}), we obtain a spliced, ideal connection γ′(p) = [A, x] ∈ B_{κ−ℓ}^{w}(X) × Sym^{ℓ}(X) by the following procedure:

• The connection A is obtained by splicing the background connection A_{0} and the connections A_{i} on S^{4} for which λ_{i} > 0, exactly as in §3.3.2.

• The multiset x ∈ Sym^{ℓ}(X) is the union of

– All points x_{i} ∈ π_{2}(p), where π_{2} is the projection from Ḡl_{κ}^{w,+}(X, U_{0}, Σ̄, λ_{0}) onto the second factor, Σ̄, for which the corresponding scale λ_{i} is zero.

– All points exp_{f_{i}}(λ_{i}φ_{n}^{−1}(y)) such that λ_{i} ≠ 0, y ∈ y_{i}, and [A_{i}, y_{i}, q_{i}] ∈ M̄_{κ_{i}}^{⋄,s}(S^{4}), where y_{i} is non-empty.

The requisite domain of the splicing and gluing maps is given by Ḡl_{E}^{+}(X, U_{0}, Σ̄, λ_{0}), where the + indicates that the pairs (x, λ) must satisfy the constraints (3.27) (with equality now allowed), essentially mimicking the definition (3.28) of Gl_{E}^{+}(X, U_{0}, Σ, λ_{0}).

The pre-splicing map (3.29) has an obvious extension to a map

γ′′ : Ḡl_{κ}^{w,+}(X, U_{0}, Σ) → B̄_{κ}^{w,∗}(X) × Met(X), (3.33)

and similarly for the splicing map γ′ and the gluing map γ, which we define in §7.5. One of the principal assertions of our main result, Theorem 1.3, is that our choice of extensions of these maps is the right one, as the resulting extensions are shown to be continuous.

4. Regularity of solutions to quasi-linear systems

Let A be a C^{∞}, orthogonal connection on the SO(3) bundle 𝔤_{E} = 𝔰𝔲(E), where E is a Hermitian, rank-two vector bundle over X. Based on the approach we gave in [21] for the PU(2) monopole equations, we give the regularity theory — which we shall need both in the present article and its sequels [14], [15], [16] — for solutions a ∈ L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}) to the quasilinear elliptic system consisting of a quasi-linear equation, with a quadratic non-linearity, and the inhomogeneous Coulomb-gauge equation,

\begin{aligned}d_{A}^{+}a + (a ∧ a)^{+} & = w, \\ d_{A}^{∗}a & = ζ, \end{aligned} (4.1)

where (ζ, w) ∈ L_{k}^{p}(𝔤_{E}) ⊕ L_{k}^{p}(Λ^{+} ⊗ 𝔤_{E}). We use the isomorphism ad : 𝔤_{E} → 𝔰𝔬(𝔤_{E}) to view sections of 𝔰𝔬(𝔤_{E}) as sections of 𝔤_{E}. A systematic treatment is given in [21, §3] and we merely summarize the main results here; some special cases appear in [8] and [24]. As in [52], [56], we use a = d_{A}^{∗}v, where v ∈ L_{2}^{2}(X, Λ^{+} ⊗ 𝔤_{E}), to rewrite the first-order quasi-linear equation for a in terms of w in (4.1) as

d_{A}^{+}d_{A}^{∗}v + (d_{A}^{∗}v ∧ d_{A}^{∗}v)^{+} = w, (4.2)

to give a second-order, quasi-linear elliptic equation for v in terms of w; the Coulomb-gauge equation reduces to d_{A}^{∗}d_{A}^{∗}v = (F_{A}^{+})^{∗}v.

Though our applications of the regularity results of this section will often be to solutions a to the anti-self-equation, F^{+}(A + a) = 0, namely equation (4.1) with w = −F_{A}^{+}, we shall also need to consider the regularity theory for more general equations, such as the various forms of the extended anti-self-dual equation (and their solutions, defining extended anti-self-dual connections) introduced in [7], [25], [54], [56] — see equations (7.4), (7.6), and (7.7), for example. These more general equations can be viewed as special cases of (4.1) or (4.2) obtained by a suitable choice of w, such as that given in equation (8.5). Therefore, by considering the slightly more general equations (4.1) and (4.2), we can address the regularity theory simultaneously for all such equations.

Remark 4.1. The precise dependence of the constants C on the connection A in this section is not something we track carefully, as we are primarily interested in local L_{k}^{2} estimates of solutions a to equations (4.1) or of solutions v to equations (4.2) when k ≥ 2 or k ≥ 3. For this range of the Sobolev indices, the constants no longer depend on simply ∥F_{A}∥_{L^{2}(X)} or ∥F_{A}^{+}∥_{L^{∞}(X)}, for example. Rather, they depend on higher-order covariant derivatives of the connection A. In general, even when A is not anti-self-dual, we can always bound L_{k+1}^{2} norms of local connection one-forms in Coulomb gauge by L_{k,A}^{2} norms of the curvature, following the method of proof for [8, Lemma 2.3.11]. If A is anti-self-dual, the dependence on A reduces to the L^{2} norm of F_{A}. Thus, throughout this section, constants depending on A can be gauge-invariantly given as constants depending on L_{k,A}^{2} norms of F_{A}.

4.1. Regularity for L_{1}^{2} solutions to the inhomogeneous Coulomb-gauge and quasilinear equations. We apply the more general results of [21, §3] — concerning regularity theory for the PU(2) monopole equations — to systems (4.1) and (4.2). Briefly, an L_{1}^{2} solution a to the anti-self-dual and Coulomb-gauge equations (4.1), with an L_{k}^{2} inhomogeneous term (with k ≥ 1) is in L_{k+1}^{2}. Thus, if the inhomogeneous term is in C^{∞} then a is in C^{∞}.

Fix an exponent 1 ≤ p ≤ ∞ and an integer k ≥ 0 such that (k + 1)p > 4. Let A be an orthogonal L_{k}^{p} connection on 𝔤_{E}. For any a in C^{∞}(Λ^{1} ⊗ 𝔤_{E}), with 1 ≤ q ≤ ∞ an exponent and ℓ ≥ an integer such that L_{k}^{p} ⊂ L_{ℓ}^{q} (see [2, Theorem 5.4]), set

∥a∥_{L_{ℓ,A}^{q}(X,g)} := (∑_{i=0}^{ℓ} ∥∇_{A}^{i}a∥_{L^{q}(X,g)}^{q})^{1/q} ,

and let L_{k}^{p}(Λ^{1} ⊗ 𝔤_{E}) be the completion of Ω^{1}(𝔤_{E}) with respect to this norm. Define the Sobolev norms and Banach-space completions of the spaces Ω^{0}(𝔤_{E}) and Ω^{+}(𝔤_{E}) in the analogous manner.

Proposition 4.2. (see [21, Proposition 3.2]) Let X be a closed, oriented, Riemannian four-manifold with metric g and let E be a Hermitian, rank-two bundle over X. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X and let 2 ≤ p < 4. Then there are positive constants ε = ε(A, p) and C = C(A, p) with the following significance. Suppose that a ∈ L_{1}^{2}(X, Λ^{1} ⊗ 𝔤_{E}) is a solution to the elliptic system (4.1) over X, where (ζ, w) is in L^{p}. If ∥a∥_{L^{4}(X)} < ε then a is in L_{1}^{p} and

∥a∥_{L_{1,A}^{p}(X)} ≤ C (∥(ζ, w)∥_{L^{p}(X)} + ∥a∥_{L^{2}(X)}) .

Proposition 4.3. (see [21, Proposition 3.3]) Continue the notation of Proposition 4.2. Let k ≥ 1 be an integer and let 2 < p < ∞. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X. Suppose that a ∈ L_{1}^{p}(X, Λ^{1} ⊗ 𝔤_{E}) is a solution to the elliptic system (4.1) over X, where (ζ, w) is in L_{k}^{2}. Then a is in L_{k+1}^{2} and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on A, k, such that Q_{k}(0, 0) = 0 and

∥a∥_{L_{k+1,A}^{2}(X)} ≤ Q_{k} (∥(ζ, w)∥_{L_{k,A}^{2}(X)}, ∥a∥_{L_{1,A}^{p}(X)}) .

In particular, if (ζ, w) is in C^{∞} then a is in C^{∞} and if (ζ, w) = 0, then

∥a∥_{L_{k+1,A}^{2}(X)} ≤ C∥a∥_{L_{1,A}^{p}(X)}.

By combining Propositions 4.2 and 4.3 we obtain the desired regularity result for L_{1}^{2} solutions to the inhomogeneous Coulomb-gauge and anti-self-dual equations:

Corollary 4.4. (see [21, Corollary 3.4]) Continue the notation of Proposition 4.2. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X. Then there is a positive constant ε = ε(A) such that the following hold. Suppose that a ∈ L_{1}^{2}(X, Λ^{1} ⊗ 𝔤_{E}), is a solution to the elliptic system (4.1) over X, where (ζ, w) is in L_{k}^{2} and ∥a∥_{L^{4}(X)} < ε and k ≥ 0 is an integer. Then a is in L_{k+1}^{2} and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on A, k, such that Q_{k}(0, 0) = 0 and

∥a∥_{L_{k+1,A}^{2}(X)} ≤ Q_{k} (∥(ζ, w)∥_{L_{k,A}^{2}(X)}, ∥a∥_{L^{2}(X)}) .

In particular, if (ζ, w) is in C^{∞} then a is in C^{∞} and if (ζ, w) = 0, then

∥a∥_{L_{k+1,A}^{2}(X)} ≤ C∥a∥_{L^{2}(X)}.

4.2. Regularity for L_{2}^{2} solutions to the inhomogeneous, second-order quasi-linear equation. It remains to consider the regularity properties of the solution v to the secondorder equation (4.2), where a = d_{A}^{+,∗}v and thus

d_{A}^{+}d_{A}^{+,∗}v = d_{A}^{+}a.

The Laplacian d_{A}^{+}d_{A}^{+,∗} is elliptic, with C^{∞} coefficients, and thus standard regularity theory implies that if a ∈ L_{k}^{p}(X, Λ^{1}⊗𝔤_{E}) then d_{A}^{+}a ∈ L_{k−1}^{p}(X, Λ^{1}⊗𝔤_{E}) and so v ∈ L_{k+1}^{p}(X, Λ^{1}⊗𝔤_{E}).

It will prove convenient to consider the following generalization of the second-order system (4.2):

d_{A}^{+}d_{A}^{∗}v + (d_{A}^{∗}v ∧ d_{A}^{∗}v)^{+} + {α, d_{A}^{∗}v} = w, (4.3)

where v ∈ L_{2}^{2}(X, Λ^{+} ⊗ 𝔤_{E}), w ∈ L_{k}^{p}(X, Λ^{+} ⊗ 𝔤_{E}), and α ∈ C^{∞}(X, Λ^{1} ⊗ 𝔤_{E}). Here,

{ , } : Γ(Λ^{1} ⊗ 𝔤_{E}) × Γ(Λ^{1} ⊗ 𝔤_{E}) → Γ(Λ^{+} ⊗ 𝔤_{E}),

denotes the linear map defined by

{α, β} := (α ∧ β)^{+} + (β ∧ α)^{+}.

Then the proofs of the regularity results of §4.1 adapt with essentially no change to give the following regularity results for equation (4.3).

Proposition 4.5. (see [21, Proposition 3.2]) Let X be a closed, oriented, Riemannian fourmanifold with metric g and let E be a Hermitian, rank-two bundle over X. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X and let 2 ≤ p < 4. Suppose α ∈ C^{∞}(X, Λ^{1} ⊗ 𝔤_{E}). Then there are positive constants ε = ε(A, α, p) and C = C(A, α, p) with the following significance. Suppose that v ∈ L_{2}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) is a solution to the elliptic system (4.3) over X, where w is in L^{p}(X, Λ^{+} ⊗ 𝔤_{E}). Further, if ∥d_{A}^{∗}v∥_{L^{4}(X)} < ε then v is in L_{2}^{p} and

∥v∥_{L_{2,A}^{p}(X)} ≤ C (∥w∥_{L^{p}(X)} + ∥v∥_{L^{2}(X)}) .

Proposition 4.6. (see [21, Proposition 3.3]) Continue the notation of Proposition 4.2. Let k ≥ 1 be an integer and let 2 < p < ∞. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X. Suppose α ∈ C^{∞}(X, Λ^{1} ⊗𝔤_{E}). Suppose that v ∈ L_{2}^{p}(X, Λ^{+} ⊗𝔤_{E}) is a solution to the elliptic system (4.3) over X, where w is in L_{k}^{2}. Then v is in L_{k+2}^{2} and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on A, α, k, such that Q_{k}(0, 0) = 0 and

∥v∥_{L_{k+2,A}^{2}(X)} ≤ Q_{k} (∥w∥_{L_{k,A}^{2}(X)}, ∥v∥_{L_{2,A}^{p}(X)}) .

In particular, if w is in C^{∞} then v is in C^{∞} and if w = 0, then

∥v∥_{L_{k+2,A}^{2}(X)} ≤ C∥v∥_{L_{2,A}^{p}(X)}.

By combining Propositions 4.5 and 4.6 we obtain the desired regularity result for L_{2}^{2} solutions to the inhomogeneous general second-order anti-self-dual equation:

Corollary 4.7. (see [21, Corollary 3.4]) Continue the notation of Proposition 4.5. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X. Suppose α ∈ C^{∞}(X, Λ^{1} ⊗ 𝔤_{E}). Then there is a positive constant ε = ε(A, α) such that the following hold. Suppose that a ∈ L_{1}^{2}(X, Λ^{1} ⊗ 𝔤_{E}), is a solution to the elliptic system (4.3) over X, where w is in L_{k}^{2} and ∥d_{A}^{∗}v∥_{L^{4}(X)} < ε and k ≥ 0 is an integer. Then v is in L_{k+2}^{2} and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on A, α, k, such that Q_{k}(0, 0) = 0 and

∥v∥_{L_{k+2,A}^{2}(X)} ≤ Q_{k} (∥w∥_{L_{k,A}^{2}(X)}, ∥v∥_{L^{2}(X)}) .

In particular, if w is in C^{∞} then v is in C^{∞} and if w = 0, then

∥v∥_{L_{k+2,A}^{2}(X)} ≤ C∥v∥_{L^{2}(X)}.

4.3. Local regularity and interior estimates for L_{1}^{2} solutions to the inhomogeneous Coulomb-gauge and quasi-linear equations. In this section we specialize the results of §4.1 to the case where the reference connection is a C^{∞} flat connection, so A = Γ on the bundle 𝔤_{E} over an open subset Ω ⊂ X.

We continue to assume that X is a closed, oriented four-manifold with C^{∞} Riemannian metric g and Hermitian, rank-two bundle E extending that on Ω ⊂ X. We then have the following local versions of Propositions 4.2 and 4.3 and Corollary 4.4:

Proposition 4.8. (see [21, Proposition 3.9]) Continue the notation of the preceding paragraph. Let Ω′ ⋐ Ω be a precompact open subset and let 2 ≤ p < 4. Then there are positive constants ε = ε(Ω, p) and C = C(Ω′, Ω, p) with the following significance. Suppose that a is an L_{1}^{2}(Ω) solution to the elliptic system (4.1) over Ω, with A = Γ and where (ζ, w) is in L^{p}(Ω). If ∥a∥_{L^{4}(Ω)} < ε then a is in L_{1}^{p}(Ω′) and

∥a∥_{L_{1,Γ}^{p}(Ω′)} ≤ C (∥(ζ, w)∥_{L^{p}(Ω)} + ∥a∥_{L^{2}(Ω)}) .

Proposition 4.9. (see [21, Proposition 3.10]) Continue the notation of Proposition 4.8. Let k ≥ 1 be an integer, and let 2 < p < ∞. Suppose that a is an L_{1}^{p}(Ω) solution to the elliptic system (4.1) over Ω with A = Γ, where (ζ, w) is in L_{k}^{2}(Ω). Then a is in L_{k+1}^{2}(Ω′) and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on k, Ω′, Ω, such that Q_{k}(0, 0) = 0 and

∥a∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ Q_{k} (∥(ζ, w)∥_{L_{k,Γ}^{2}(Ω)}, ∥a∥_{L_{1,Γ}^{p}(Ω)}) .

If (ζ, w) is in C^{∞}(Ω) then a is in C^{∞}(Ω′) and if (ζ, w) = 0, then

∥a∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ C∥a∥_{L_{1,Γ}^{p}(Ω)}.

Corollary 4.10. (see [21, Corollary 3.11]) Continue the notation of Proposition 4.9. Then there is a positive constant ε = ε(Ω) with the following significance. Suppose that a is an L_{1}^{2}(Ω) solution to the elliptic system (4.1) over Ω with A = Γ, where (ζ, w) is in L_{k}^{2}(Ω) and ∥a∥_{L^{4}(Ω)} < ε. Then a is in L_{k+1}^{2}(Ω′) and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on k, Ω′, Ω, such that Q_{k}(0, 0) = 0 and

∥a∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ Q_{k} (∥(ζ, w)∥_{L_{k,Γ}^{2}(Ω)}, ∥a∥_{L^{2}(Ω)}) .

If (ζ, w) is in C^{∞}(Ω) then a is in C^{∞}(Ω′) and if (ζ, w) = 0, then

∥a∥_{L_{k+1,Γ}^{2}(Ω′)} ≤ C∥a∥_{L^{2}(Ω)}.

Corollary 4.10 thus yields a sharp local elliptic regularity result for L_{1}^{2} anti-self-dual connections A which are given to us in Coulomb gauge relative to Γ.

Proposition 4.11. (see [21, Proposition 3.12]) Continue the notation of Corollary 4.10. Then there is a positive constant ε = ε(Ω) and, if k ≥ 1 is an integer, there is a positive constant C = C(Ω′, Ω, k) with the following significance. Suppose that A is an L_{1}^{2} solution to the g-anti-self-dual equation over Ω, which is in Coulomb gauge over Ω relative to Γ, so d_{Γ}^{∗}(A − Γ) = 0, and obeys ∥A − Γ∥_{L^{4}(Ω)} < ε. Then A − Γ is in C^{∞}(Ω′) and for any k ≥ 1,

∥A − Γ∥_{L_{k,Γ}^{2}(Ω′)} ≤ C∥A − Γ∥_{L^{2}(Ω)}.

4.4. Local regularity and interior estimates for L_{2}^{2} solutions to the general, inhomogeneous, second-order quasi-linear equation. In this section we specialize the results of §4.2 to the case of an open subset Ω ⊂ X.

We continue to assume that X is a closed, oriented four-manifold with C^{∞} Riemannian metric g and Hermitian, rank-two bundle E extending that on Ω ⊂ X. We then have the following local versions of Propositions 4.5 and 4.6 and Corollary 4.7:

Proposition 4.12. (see [21, Proposition 3.9]) Continue the notation of the preceding paragraph. Suppose α ∈ C^{∞}(X, Λ^{1} ⊗ 𝔤_{E}). Let Ω′ ⋐ Ω be a precompact open subset and let 2 ≤ p < 4. Then there are positive constants ε = ε(A, α, Ω, p) and C = C(A, α, Ω′, Ω, p) with the following significance. Suppose that v is an L_{2}^{2}(Ω) solution to the elliptic system (4.3) over Ω, where w is in L^{p}(Ω). If ∥d_{A}^{∗}v∥_{L^{4}(Ω)} < ε then v is in L_{2}^{p}(Ω′) and

∥v∥_{L_{2,A}^{p}(Ω′)} ≤ C (∥w∥_{L^{p}(Ω)} + ∥v∥_{L^{2}(Ω)}) .

Proposition 4.13. (see [21, Proposition 3.10]) Continue the notation of Proposition 4.12. Let k ≥ 1 be an integer, and let 2 < p < ∞. Suppose that v is an L_{2}^{p}(Ω) solution to the elliptic system (4.3) over Ω, where w is in L_{k}^{2}(Ω). Then v is in L_{k+2}^{2}(Ω′) and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on A, α, k, Ω′, Ω, such that Q_{k}(0, 0) = 0 and

∥v∥_{L_{k+2,A}^{2}(Ω′)} ≤ Q_{k} (∥w∥_{L_{k,A}^{2}(Ω)}, ∥v∥_{L_{2,A}^{p}(Ω)}) .

If w is in C^{∞}(Ω) then v is in C^{∞}(Ω′) and if w = 0, then

∥v∥_{L_{k+2,A}^{2}(Ω′)} ≤ C∥v∥_{L_{2,A}^{p}(Ω)}.

Corollary 4.14. (see [21, Corollary 3.11]) Continue the notation of Proposition 4.13. Then there is a positive constant ε = ε(A, α, Ω) with the following significance. Suppose that v is an L_{2}^{2}(Ω) solution to the elliptic system (4.3) over Ω, where w is in L_{k}^{2}(Ω) and ∥d_{A}^{∗}v∥_{L^{4}(Ω)} < ε. Then v is in L_{k+2}^{2}(Ω′) and there is a universal polynomial Q_{k}(x, y), with positive real coefficients, depending at most on A, α, k, Ω′, Ω, such that Q_{k}(0, 0) = 0 and

∥v∥_{L_{k+2,A}^{2}(Ω′)} ≤ Q_{k} (∥w∥_{L_{k,A}^{2}(Ω)}, ∥v∥_{L^{2}(Ω)}) .

If w is in C^{∞}(Ω) then v is in C^{∞}(Ω′) and if w = 0, then

∥v∥_{L_{k+2,A}^{2}(Ω′)} ≤ C∥v∥_{L^{2}(Ω)}.

4.5. Estimates for extended anti-self-dual connections in a good local gauge. It remains to combine the local regularity results and estimates of §4.3, for extended anti-self-dual connections A where the connection A is assumed to be in Coulomb gauge relative to the product SO(3) connection Γ, with Uhlenbeck’s local, Coulomb gauge-fixing theorem. We then obtain regularity results and estimates for anti-self-dual connections A with small curvature F_{A} (see Theorem 2.3.8 and Proposition 4.4.10 in [8]).

In order to apply Corollary 4.10 we need Uhlenbeck’s Coulomb gauge-fixing result [60, Theorem 2.1 & Corollary 2.2]. Let B (respectively, B̄) be the open (respectively, closed) unit ball centered at the origin in ℝ^{4} and let G be a compact Lie group. In order to provide universal constants we assume ℝ^{4} has its standard metric, though the results of this subsection naturally hold for any C^{∞} Riemannian metric, with comparable constants for metrics which are suitably close.

Theorem 4.15. There are positive constants c and ε with the following significance. If 2 ≤ p < 4 is a constant and A ∈ L_{1}^{p}(B, Λ^{1}⊗𝔤)∩L_{1}^{p}(∂B, Λ^{1}⊗𝔤) is a connection matrix whose curvature satisfies ∥F_{A}∥_{L^{p}(B)} < ε, then there is a gauge transformation u ∈ L_{2}^{p}(B, G) ∩ L_{2}^{p}(∂B, G) such that u(A) := uAu^{−1} − (du)u^{−1} satisfies

(1)  d^{∗}u(A) = 0 on B,
(2)  \frac{∂}{∂r} ⌟u(A) = 0 on ∂B,
(3)  ∥u(A)∥_{L_{1}^{p}(B)} ≤ c∥F_{A}∥_{L^{p}(B)}.

If A is in L_{k}^{p}(B), for k ≥ 2, then u is in L_{k+1}^{p}(B). The gauge transformation u is unique up to multiplication by a constant element of G.

Remark 4.16. 1. The L_{2}^{p} gauge transformation u is obtained by solving the secondorder quasi-linear elliptic equation d^{∗}A^{u} = 0, that is

d^{∗}du = − ∗ (duu^{−1} ∧ ∗du) − d^{∗}Au + ∗(∗A ∧ du) − ∗(duu^{−1} ∧ ∗Au),

given A in L_{1}^{p}. For any open subset U ⋐ B, there is a positive constant ε_{0}(U) such that if A in L_{2}^{p}(B) satisfies ∥A∥_{L_{2}^{p}(B)} < ε_{0} then u is in L_{3}^{p}(U) by the implicit function theorem. In particular, if A is in C^{∞}(B) then standard elliptic theory and the Sobolev multiplication theorem imply that u is in C^{∞}(U) [8, Proposition 2.3.4].

2. Uhlenbeck’s methods [60] imply that the gauge transformation u_{A} depends analytically on the connection A.

3. If G is abelian then the requirement that ∥F_{A}∥_{L^{p}(B)} < ε can be omitted.

It is often useful to rephrase Theorem 4.15 in two other slightly different ways. Suppose A is an L_{k}^{2} connection on a C^{∞} principal G bundle P over B with k ≥ 2 and ∥F_{A}∥_{L^{2}(B)} < ε. Then the assertions of Theorem 4.15 are equivalent to each of the following:

• There is an L_{k+1}^{2} trivialization τ : P → B×G such that (i) d_{Γ}^{∗}(τ(A)−Γ) = 0, where Γ is the product connection on B ×G, (ii) \frac{∂}{∂r} ⌟(τ(A)−Γ) = 0, and (iii) ∥(τ(A)−Γ)∥_{L_{1}^{2}(B)} ≤ c∥F_{A}∥_{L^{2}(B)}.

• There is an L_{k+1}^{2} flat connection Γ on P such that (i) d_{Γ}^{∗}(A−Γ) = 0, (ii) \frac{∂}{∂r} ⌟(A−Γ) = 0, and (iii) ∥(A − Γ)∥_{L_{1}^{2}(B)} ≤ c∥F_{A}∥_{L^{2}(B)}, and an L_{k+1}^{2} trivialization P |_{B} ≃ B × G taking Γ to the product connection.

We can now combine Theorem 4.15 with Corollary 4.10 or Proposition 4.11 to give (see [8, Theorem 2.3.8], [24, Proposition 8.3]):

Theorem 4.17. Let B ⊂ ℝ^{4} be the open unit ball with center at the origin, let U ⋐ B be an open subset, and let Γ be the product connection on B × SO(3). Then there is a positive constant ε and if k ≥ 1 is an integer, there is a positive constant C(k, U) with the following significance. Suppose that A is an L_{1}^{2} solution to the extended anti-self-dual connection on 𝔤_{E} over B, namely

F_{A}^{+} = w,

for some w ∈ L_{k−1}^{2}(B, Λ^{+} ⊗ 𝔤_{E}), and that the curvature of the SO(3) connection matrix A obeys

∥F_{A}∥_{L^{2}(B)} < ε.

Then there is an L_{k+1}^{2} gauge transformation u : B → SU(2) such that u(A) − Γ is in L_{k}^{2}(U) with d^{∗}(u(A) − Γ) = 0 over B and

∥u(A) − Γ∥_{L_{k,Γ}^{2}(U)} ≤ Q_{k} (∥u(w)∥_{L_{k−1,Γ}^{2}(B)}, ∥F_{A}∥_{L^{2}(B)}) .

If u(w) is in C^{∞}(B), then u(A) − Γ is in C^{∞}(U). If w = 0, so that A is an anti-self-dual connection, then

∥u(A) − Γ∥_{L_{k,Γ}^{2}(U)} ≤ C∥F_{A}∥_{L^{2}(B)}.

Remark 4.18. The L_{k}^{2} estimates for A obtained from Theorem 4.17 are uniform over all metrics in a small C^{k} neighborhood of a fixed metric [24, p. 120].

Again, it is often useful to rephrase Theorem 4.17 in the two other slightly different ways. Suppose ℓ ≥ 1 and that A is an L_{k}^{2} anti-self-dual connection on 𝔤_{E} over the unit ball B ⊂ ℝ^{4} with k ≥ max{2, ℓ}, ∥F_{A}∥_{L^{2}(B)} < ε and U ⋐ B. Then the assertions of Theorem 4.17 are equivalent to each of the following:

• There is a C^{∞} trivialization τ : 𝔤_{E}|_{B} → B×𝔰𝔲(2) and a L_{k+1}^{2} determinant-one, unitary bundle automorphism u of E|_{B} such that, with respect to the product connection Γ on B × 𝔰𝔲(2), we have (i) d_{Γ}^{∗}(τu(A) − Γ) = 0, and (ii) ∥(τu(A) − Γ, τuΦ)∥_{L_{ℓ,Γ}^{2}(U)} ≤ C∥F_{A}∥_{L^{2}(B)}.

• There is an L_{k+1}^{2} flat connection Γ on 𝔤_{E}|_{B} such that (i) d_{Γ}^{∗}(A − Γ) = 0, and (ii) ∥A − Γ∥_{L_{ℓ,Γ}^{2}(U)} ≤ c∥F_{A}∥_{L^{2}(B)}, and an L_{k+1}^{2} trivialization 𝔤_{E}|_{B} ≃ B × 𝔰𝔲(2) taking Γ to the product connection.

We will need interior estimates for anti-self-dual connections in a good local gauge over more general simply-connected regions than the open balls considered in Theorem 4.17. Specifically, recall that a domain Ω ⊂ X is strongly simply-connected if it has an open covering by balls D_{1}, …, D_{m} (not necessarily geodesic) such that for 1 ≤ r ≤ m the intersection D_{r} ∩ (D_{1} ∪ ⋯ ∪ D_{r−1}) is connected. We recall (see [8, Proposition 2.2.3] or [33, Proposition I.2.6]):

Proposition 4.19. If Γ is a C^{∞} flat connection on a principal G bundle P over a simplyconnected manifold Ω, then there is a C^{∞} isomorphism P ≃ Ω × G taking Γ to the product connection on Ω × G.

More generally, if A is C^{∞} connection on a G bundle P over a simply-connected manifoldwith-boundary Ω̄ = Ω∪∂Ω with L^{p}-small curvature (with p > 2), then Uhlenbeck’s theorem implies that A is L_{2}^{p}-gauge equivalent to a connection which is L_{1}^{p}-close to an L_{1}^{p} flat connection on P (see [61, Corollary 4.3] or [8, p. 163]). We recall the following a priori interior estimate (see also [21, Proposition 3.18] and [8, Proposition 4.4.10]).

Proposition 4.20. Let X be a closed, oriented, Riemannian four-manifold and let Ω ⊂ X be a strongly simply-connected open subset. Then there is a positive constant ε(Ω) with the following significance. For Ω′ ⋐ Ω a precompact open subset and an integer ℓ ≥ 1, there is a constant C(ℓ, Ω′, Ω) such that the following holds. Suppose A is an L_{k}^{2} extended anti-self-dual connection on 𝔤_{E} over Ω, namely

F_{A}^{+} = w,

for some w ∈ L_{k−1}^{2}(Ω, Λ^{+} ⊗ 𝔤_{E}), with k ≥ max{2, ℓ} such that

∥F_{A}∥_{L^{2}(Ω)} < ε.

Then there is an L_{k+1}^{2} flat connection Γ on 𝔤_{E}|_{Ω′} such that

∥A − Γ∥_{L_{ℓ,Γ}^{2}(Ω′)} ≤ Q_{k} (∥w∥_{L_{k−1,Γ}^{2}(Ω)}, ∥F_{A}∥_{L^{2}(Ω)}) ,

and an L_{k+1}^{2} trivialization 𝔤_{E}|_{Ω′} ≃ Ω′ × 𝔰𝔲(2) taking Γ to the product connection. If w is in C^{∞}(Ω), then A, Γ are in C^{∞}(Ω′). If w = 0, so that A is an anti-self-dual connection, then

∥A − Γ∥_{L_{ℓ,Γ}^{2}(Ω′)} ≤ C∥F_{A}∥_{L^{2}(Ω)}.

For some local patching arguments over simply-connected open regions Y ⊂ X it is very useful to be able to lift gauge transformations in G_{𝔰𝔲(E)}(Y) to gauge transformations in G_{E}(Y). The following result tells us that this is always possible when Y is simply connected; it is an extension of Theorem IV.3.1 in [44] from SU(2) to U(2) bundles.

Proposition 4.21. [21, Proposition 2.16] Let E be a rank-two, Hermitian bundle over a connected manifold X. Then there is an exact sequence ^{5}

1 → {±id_{E}} → G_{E} → G_{𝔰𝔲(E)} → H^{1}(X; ℤ/2ℤ).

4.6. Regularity of Taubes’ solutions to extended anti-self-dual equations. Finally, we come to one of the main results of this section.

Proposition 4.22. Let X be a closed, oriented, C^{∞} four-manifold with C^{∞} Riemannian metric g and let E be a Hermitian, rank-two bundle over X. Let A be a C^{∞} connection on the C^{∞} bundle 𝔤_{E} over X. Suppose that a ∈ L_{1}^{2}(X, Λ^{1} ⊗ 𝔤_{E}) is a solution to an extended anti-self-dual equation over X,

F^{+,g}(A + a) = w,

where a = d_{A}^{+,∗}v, for some v ∈ (C^{0} ∩ L_{2}^{2})(X, Λ^{+} ⊗ 𝔤_{E}), and w ∈ C^{∞}(X, Λ^{+} ⊗ 𝔤_{E}). Then a is in C^{∞}(X, Λ^{1} ⊗ 𝔤_{E}) and v ∈ C^{∞}(X, Λ^{+} ⊗ 𝔤_{E}).

Remark 4.23. The reader may wonder why we cannot apply Corollary 4.4 to the resulting elliptic system (4.4) and deduce straightaway that the solution a is in C^{∞}. The difficulty, of course, is that the hypotheses of Corollary 4.4 are not necessarily satisfied due to the possibly unfavorable dependence of the constant ε(A) on the connection A. For example, by combining results of §5 and §7, we can show that ∥a∥_{L^{4}(X)} ≤ Cλ when A_{λ} is a spliced connection (produced by the alogorithm of §3), λ is a parameter which can be made arbitrarily small, and C is an essentially universal constant, depending at most on ∥F_{A_{λ}}^{+}∥_{L^{∞}(X)} and ∥F_{A_{λ}}∥_{L^{2}}: in particular, C is independent of λ. Now, as we make λ smaller to try to satisfy the constraint of Corollary 4.4, we will in general find that ε(A_{λ}) also becomes smaller (as it will, for example, if ε(A_{λ}) depends on an L_{ℓ}^{p} norm of F_{A_{λ}} with p > 2, ℓ = 0 or ℓ ≥ 1, p = 2). Thus, as we do not necessarily have a lower bound for ε(A) which is uniform with respect to A, we are left ‘chasing our tail’ and Corollary 4.4 will not apply.

5It can also be shown that the map G_{𝔰𝔲(E)} → H^{1}(X; ℤ/2ℤ) is surjective.

Proof of Proposition 4.22. Since a = d_{A}^{+,∗}v, the extended anti-self-dual and Coulomb-gauge equations become

\begin{aligned}d_{A}^{+}a + (a ∧ a)^{+} & = w − F_{A}^{+}, \\ d_{A}^{∗}a & = (F_{A}^{+})^{∗}v, \end{aligned} (4.4)

where F_{A}^{+} ∈ C^{∞}(Λ^{+} ⊗ 𝔤_{E}) and (F_{A}^{+})^{∗}v ∈ (C^{0} ∩ L_{2}^{2})(Λ^{+} ⊗ 𝔤_{E}). Observe that

(ζ, w′) := ((F_{A}^{+})^{∗}v, w − F_{A}^{+}) ∈ L_{2}^{2}(𝔤_{E}) ⊕ L_{2}^{2}(Λ^{+} ⊗ 𝔤_{E}).

From Remark 4.23 we note that we cannot take the apparently obvious route and apply Corollary 4.4. Instead, we fix a point x ∈ X and a geodesic ball B(x, δ) centered at x, fix a C^{∞} trivialization 𝔤_{E}|_{B(x,δ)} ≃ B(x, δ)×𝔰𝔲(2), and let Γ be the resulting product connection. Then

\begin{aligned}d^{+}b + (b ∧ b)^{+} & = w − F_{Γ}^{+} = w, \\ d^{∗}b & = ∗((A − Γ) ∧ ∗a) + d^{∗}(A − Γ) + (F_{A}^{+})^{∗}v, \end{aligned} (4.5)

where b := A − Γ + a ∈ L_{1}^{2}(B(x, δ), Λ^{1} ⊗ 𝔤_{E}) and the first equation is the expansion of F^{+}(Γ + b) = w. In particular, the right-hand side of equation (4.5) is in L_{1}^{2}(B(x, δ)).

In Corollary 4.10, let Ω = B ⊂ ℝ^{4} be the unit ball, let ε = ε(Ω) be the corresponding positive constant, and let Ω′ = B′ be a slightly smaller concentric ball. Since a ∈ L^{4}(X, Λ^{1}⊗ 𝔤_{E}), we know that ∥a∥_{L^{4}(U)} → 0 as Vol(U) → 0, where U is any measurable subset of X. In particular, for a small enough ball B(x, δ), we have ∥b∥_{L^{4}(B(x,δ))} < ε. We now rescale the metric on the ball B(x, δ) to give a ball B̃ with unit radius, center x, and metric which is approximately flat. Note that the equations (4.5) are scale-equivariant and that

∥b∥_{L^{4}(B̃)} = ∥b∥_{L^{4}(B(x,δ))} < ε,

by the scale-invariance of the L^{4} norm on one-forms. Corollary 4.10 now implies that b — and thus a — is in L_{2}^{2} on the slightly smaller rescaled ball B̃′. Since d_{A}^{+}d_{A}^{+,∗}v = d_{A}^{+}a is now in L_{1}^{2}(B̃′), we have v ∈ L_{3}^{2}(B̃′) by standard elliptic regularity theory for a Laplacian, d_{A}^{+}d_{A}^{+,∗}, with C^{∞} coefficients [32]. Hence, the right-hand side of equation (4.5) is in L_{2}^{2}(B̃′). Repeating the above argument shows that a is in C^{∞} on the ball B̃_{1/2} ⊂ B̃ with center x and radius \frac{1}{2}. Hence, a is in C^{∞} on B(x, δ/2) and so, as the point x ∈ X was arbitrary, we see that a ∈ C^{∞}(X, Λ^{1} ⊗ 𝔤_{E}) and v ∈ C^{∞}(X, Λ^{+} ⊗ 𝔤_{E}).

5. Decay estimates for curvature and connections in Coulomb and radial gauges

Our splicing and unsplicing constructions for connections requires estimates for local connection one-forms in suitable gauges, usually Coulomb or radial, over small balls, over their complements, and over annuli. The most useful estimates bound Sobolev or Hölder norms of local connection one-forms in terms of the L^{2} norms of their curvatures and the radii defining the given regions, with estimate constants which are essentially universal. These are the kinds of estimates we derive in this section. Some of these bounds can be derived from the curvature decay estimates due, in various forms, to Donaldson, Groisser-Parker, Råde, Taubes, and Uhlenbeck. Perhaps the most useful and powerful statement is due to Donaldson, Groisser-Parker and Råde; see the discussion in [14]. However, we prefer to use methods which are as simple as possible. Furthermore, our chosen methods have the the following advantages:

• They extend easily to the case of solutions to extended anti-self-dual equations, such as equations (7.1) or (7.3).

• They extend easily to the case of PU(2) monopoles [18]. In contrast, the proofs of the elegant, pointwise decay estimates of Donaldson, Groisser-Parker, and Råde — for the curvature of anti-self-dual or Yang-Mills connections — requires a delicate application of the Chern-Simons functional and considerably more work is needed to extend them to the case of PU(2) monopoles, though decay estimates of the desired type for Seiberg-Witten monopoles are described in [47]. It is likely that their estimates can be extended to cover the case of extended anti-self-dual connections, but again some work is required to show this. The disadvantage of our approach is that we must separately consider the cases of (i) balls, (ii) complements of balls in ℝ^{4}, and (iii) annuli, while the decay estimate theorem of Donaldson, Groisser-Parker, and Råde would give a more uniform treatment of these different regions.

In all cases, the estimates of this section continue to hold in situations where the Riemannian metric is uniformly C^{r} close enough to the standard metric. Thus, for the sake of notational simplicity, we shall assume that our connections are defined on regions in ℝ^{4} equipped with its standard Riemannian metric g. We let B(r) denote the open ball {x ∈ ℝ^{4} : |x| < r} of radius r and center at the origin and let B̄(r) denote the closed ball {x ∈ ℝ^{4} : |x| ≤ r}.

5.1. Estimates for anti-self-dual connections over balls. The basic case is that of ball in ℝ^{4}: estimates for connections over the complements of balls and over annuli follow from this. Theorem 4.17 yields an interior L^{∞} bound for the curvature F_{A} of an anti-self-dual connection over a ball in terms of the L^{2} norm of F_{A}; compare Proposition 8.3 in [24] and Lemma 9.1 in [55].

Lemma 5.1. There are positive constants ε and, given a constant 1 ≤ p ≤ ∞ and an integer k ≥ 0, a positive constant c(k, p) with the following significance. Let G be a compact Lie group. Suppose r_{1} > 0 is a constant and that A is an anti-self-dual connection on the product bundle P = B(r_{1}) × G, where B(r_{1}) is a ball centered at the origin in ℝ^{4}, with its Euclidean metric, such that

∥F_{A}∥_{L^{2}(B(r_{1}))} < ε,

then

∥∇_{Γ}^{k}F_{A}∥_{L^{p}(B(r_{1}/2))} ≤ c(k, p)r_{1}^{(4/p)−k−2}∥F_{A}∥_{L^{2}(B(r_{1}))}.

Moreover, if Γ is the product connection on ℝ^{4} × G and d_{Γ}^{∗}(A − Γ) = 0, then

∥∇_{Γ}^{k}(A − Γ)∥_{L^{p}(B(r_{1}/2))} ≤ c(k, p)r_{1}^{(4/p)−k−1}∥F_{A}∥_{L^{2}(B(r_{1}))}.

Finally, the estimates for F_{A} also hold when ∇_{Γ} is replaced by ∇_{A}.

Proof. We define a rescaled metric g̃ on ℝ^{4} by setting g̃_{μν} = r_{1}^{−2}g_{μν}, so g̃^{μν} = r_{1}^{2}g^{μν} and B_{g}(r_{1}) = Bg̃(1). Then,

\begin{aligned}|∇_{Γ}^{k}(A − Γ)|_{g} & = r_{1}^{−k−1}|∇_{Γ}^{k}(A − Γ)|g̃, \\ |∇_{A}^{k}F_{A}|_{g} & = r_{1}^{−k−2}|∇_{A}^{k}F_{A}|g̃. \end{aligned} (5.1)

Since ∥F_{A}∥_{L^{2}(B(1),g̃)} < ε and ε is sufficiently small, Theorem 4.15 provides a gauge transformation u over Bg̃(1) such that d_{Γ}^{∗}(u(A) − Γ) = 0, so without loss of generality we may assume that A is already in Coulomb gauge, as we suppose in the second assertion of the Lemma. Theorem 4.17, noting that its proof extends from p = 2 to the general case in a straightforward manner, then yields

∥A − Γ∥_{L_{k,Γ}^{p}(B(1/2),g̃)} ≤ c(k, p)∥F_{A}∥_{L^{2}(B(1),g̃)},

and the desired Sobolev L_{k}^{p} estimates follow, since

∥∇_{Γ}^{k}(A − Γ)∥_{L^{p}(B(r_{1}/2),g)} = r_{1}^{(4/p)−k−1}∥∇_{Γ}^{k}(A − Γ)|_{L^{2}(B(1/2),g̃)},
∥∇_{A}^{k}F_{A}∥_{L^{p}(B(r_{1}/2),g)} = r_{1}^{(4/p)−k−2}∥∇_{A}^{k}F_{A}∥_{L^{2}(B(1/2),g̃)}.

This completes the proof.

In our treatment of ‘ungluing’ in [14] we shall need L_{1}^{2} estimates for covariant derivatives of radial-gauge connection one-forms, while L^{p} estimates are needed throughout our article. Therefore, suppose that the connection one-form A − Γ is in radial gauge with respect to the origin, x^{μ}A_{μ}(x) = 0, so [59]

A_{ν}(x) = ∫_{0}^{1} tx^{μ}(F_{A})_{μν}(ty) dt, (5.2)

where A_{μ} = (A − Γ)_{μ}. We then have the following estimates for the local connection one-forms and their derivatives (compare Taubes’ Lemmas 9.1 and 9.2 in [55]):

Lemma 5.2. Continue the hypotheses and notation of Lemma 5.1. Suppose that the connection one-form A − Γ is in radial gauge with respect to the origin. Then, for any 1 ≤ p ≤ ∞,

∥∇_{Γ}^{k}(A − Γ)∥_{L^{p}(B(r_{1}/2))} ≤ c(k, p)r_{1}^{(4/p)−k−1}∥F_{A}∥_{L^{2}(B(r_{1}))}.

Proof. Denote A − Γ simply by A. We confine our attention to the case k = 0, 1 (which are the only ones we shall need) and leave the general case to the reader. From equation (5.2) we have

(5.3)  |A|(x) ≤ |x| · |F_{A}|(x), x ∈ ℝ^{4},
|∇_{Γ}A|(x) ≤ |x| · |∇_{Γ}F_{A}|(x) + |F_{A}|(x).

Lemma 5.1 then yields the following Sobolev estimates. First,

∥A∥_{L^{p}(B(r_{1}/2)} ≤ cr_{1}^{(4/p)−1}∥F_{A}∥_{L^{2}(B(r_{1}))},

which gives the desired bound when k = 0. For k = 1, we have

∥∇_{Γ}A∥_{L^{p}(B(r_{1}/2)} ≤ r_{1}∥∇_{Γ}F_{A}∥_{L^{p}(B(r_{1}/2))} + ∥F_{A}∥_{L^{p}(B(r_{1}/2))}
≤ cr_{1}^{4/p}(r_{1} · r_{1}^{−3} + r_{1}^{−2})∥F_{A}∥_{L^{2}(B(r_{1}))}
= cr_{1}^{(4/p)−2}∥F_{A}∥_{L^{2}(B(r_{1}))}.

The cases k ≥ 2 are similar.

Remark 5.3. The estimates in Lemmas 5.1 and 5.2 agree in the case k = 0 for F_{A} with that obtained by integrating the decay estimate of Donaldson et al. when r_{0} = 0. The estimates for F_{A} and A − Γ agree with those obtained by integrating the pointwise decay estimates of [55, Lemma 9.1].

5.2. Estimates for anti-self-dual connections over the complement of a ball. We now consider the case where A is an anti-self-dual connection with small curvature on the complement of a ball in ℝ^{4}.

Our first task is to note that given a C^{∞} anti-self-dual connection A on ℝ^{4} − B(r_{0}), with finite energy, there is no loss in generality — thanks to the removable singularities theorem of K. Uhlenbeck [8, Theorem 4.4.12], [59, Theorem 4.1] — if we assume that A is a C^{∞} connection on S^{4} − φ_{n}(B(r_{0})), where the origin of ℝ^{4} is identified with the north pole of S^{4}.

Theorem 5.4. Let A be a C^{∞} anti-self-dual connection on a G bundle P over the punctured unit ball B \ {0} in ℝ^{4}, with its Euclidean metric. If

∥F_{A}∥_{L^{2}(B)} < ∞,

there is a C^{∞} anti-self-dual connection Ā on a C^{∞} bundle \overline{P} over B and a C^{∞} bundle isomorphism u : \overline{P} |_{B\\{0\}} → P |_{B\\{0\}} such that u(Ā) = A over B \ \{0\}.

We now consider the problem of estimating the L_{k}^{p} norms of F_{A} and A − Γ (in radial gauge) on ℝ^{4} − B(r_{0}).

Lemma 5.5. There are positive constants ε and, given an integer k ≥ 0 and a constant 1 ≤ p ≤ ∞, a positive constant c(k, p) with the following significance. Let G be a compact Lie group. Suppose r_{0} > 0 is a constant and that A is an anti-self-dual connection on the product bundle P = (ℝ^{4} − B(r_{0})) × G, where B(r_{0}) is a ball centered at the origin in ℝ^{4}, with its Euclidean metric, such that

∥F_{A}∥_{L^{2}(ℝ^{4}−B(r_{0}))} < ε,

then, provided k + 2 − 4/p ≥ 0,

∥∇_{A}^{k}F_{A}∥_{L^{p}(ℝ^{4}−B(2r_{0}))} ≤ c(k, p)r_{0}^{(4/p)−k−2}∥F_{A}∥_{L^{2}(ℝ^{4}−B(r_{0}))}.

Moreover, if Γ is the product connection on (ℝ^{4} − B(r_{0})) × G and A − Γ is in radial gauge with respect to ∞ then, for k + 1 − 4/p ≥ 0,

∥∇_{Γ}^{k}(A − Γ)∥_{L^{p}(ℝ^{4}−B(2r_{0}))} ≤ c(k, p)r_{0}^{(4/p)−k−1}∥F_{A}∥_{L^{2}(ℝ^{4}−B(r_{0}))}.

Proof. Identifying ℝ^{4} with the quaternions ℍ, we recall that y = ι^{∗}(y) = φ_{s}^{−1} ∘ φ_{n}(x) = x^{−1} for x ∈ ℝ^{4} − {0}, where φ_{s} : S^{4} − {n} → ℝ^{4} and φ_{n} : S^{4} − {s} → ℝ^{4} are stereographic projections. The inversion map is conformal, so ι^{∗}A is an anti-self-dual connection on the punctured ball B̄(1/r_{0}) − {0}, with curvature bound

∥F_{ι^{∗}A}∥_{L^{2}(B(1/r_{0}))} = ∥F_{A}∥_{L^{2}(ℝ^{4}−B(r_{0}))} < ε.

As ∥F_{ι^{∗}A}∥_{L^{2}(B(1/r_{0}))} < ∞, Theorem 5.4 implies that there is a C^{∞} gauge transformation u of B̄(1/r_{0}) × G such that the connection one-form u(ι^{∗}A) extends to a C^{∞} connection over the ball B(1/r_{0}). Thus, we may consider A to be a C^{∞} connection on the product bundle P = (S^{4} − φ_{n}(B(r_{0}))) × G, with product connection Γ.

From the identity |dx| = |y|^{−2}|dy|, we obtain

|φ_{n}^{∗}∇_{A}^{k}F_{A}|(x) = |y|^{−2(k+2)} · |φ_{s}^{∗}∇_{A}^{k}F_{A}|(y),
|φ_{n}^{∗}∇_{Γ}^{k}(A − Γ)|(x) = |y|^{−2(k+1)} · |φ_{s}^{∗}∇_{Γ}^{k}(A − Γ)|(y).

Combining these observations and noting that d^{4}x = |y|^{−8}d^{4}y, we see that (for 2(k + 2) − 8/p ≥ 0 and 2(k + 1) − 8/p ≥ 0, respectively)

∥φ_{n}^{∗}∇_{A}^{k}F_{A}∥_{L^{p}(ℝ^{4}−B(2r_{0}))} = ∥|y|^{2(k+2)−8/p}φ_{s}^{∗}∇_{A}^{k}F_{A}∥_{L^{p}(B(1/2r_{0}))}
≤ (2r_{0})^{−2(k+2)+8/p}∥φ_{s}^{∗}∇_{A}^{k}F_{A}∥_{L^{p}(B(1/2r_{0}))},
∥φ_{n}^{∗}∇_{Γ}^{k}(A − Γ)∥_{L^{p}(ℝ^{4}−B(2r_{0}))} = ∥|y|^{2(k+1)−8/p}φ_{s}^{∗}∇_{Γ}^{k}(A − Γ)∥_{L^{p}(B(1/2r_{0}))}
≤ (2r_{0})^{−2(k+1)+8/p}∥φ_{s}^{∗}∇_{Γ}^{k}(A − Γ)∥_{L^{p}(B(1/2r_{0}))}.

Applying Lemmas 5.1 and 5.2 to the connection φ_{s}^{∗}A on B(1/r_{0}), in radial gauge with respect to the origin, we see that

∥φ_{s}^{∗}∇_{A}^{k}F_{A}∥_{L^{p}(B(1/2r_{0}))} ≤ c(k, p)r_{0}^{−(4/p)+k+2}∥F_{A}∥_{L^{2}(B(1/r_{0}))},
∥φ_{s}^{∗}∇_{Γ}^{k}(A − Γ)∥_{L^{p}(B(1/2r_{0}))} ≤ c(k, p)r_{0}^{−(4/p)+k+1}∥F_{A}∥_{L^{2}(B(1/r_{0}))}.

Combining the preceding two pairs of estimates then yields the desired bounds.

Remark 5.6. The estimates in Lemma 5.5 agree in the case k = 0 for F_{A} with that obtained by integrating the decay estimate of Donaldson et al. when r_{1} = ∞. The estimates for F_{A} and A − Γ agree with those obtained by integrating the pointwise decay estimates of [55, Lemma 9.2].

5.3. Estimates for anti-self-dual connections over annuli. The case of an annulus requires a slightly more sophisticated approach and a complete treatment — where both inner and outer radius can be specified independently — requires the decay estimate of Donaldson et al. For positive constants r_{0} < r_{1}, let

Ω(r_{0}, r_{1}) := {x ∈ ℝ^{4} : r_{0} < |x| < r_{1}}.

We then have the following consequence of Proposition 4.20.

Lemma 5.7. There are positive constants ε and, given an integer k ≥ 0 and a constant 1 ≤ p ≤ ∞, a positive constant c(k, p) with the following significance. Let G be a compact Lie group. Suppose r_{1} > 0 is a constant and that A is an C^{∞} anti-self-dual connection on a G bundle P over the annulus Ω = Ω(\frac{1}{4}r_{1}, 4r_{1}) in ℝ^{4}, with its Euclidean metric, such that

∥F_{A}∥_{L^{2}(Ω( \frac{1}{4} r_{1},4r_{1}))} < ε.

Then there is an C^{∞} flat connection on P |_{Ω′} such that

∥∇_{Γ}^{k}(A − Γ)∥_{L^{p}(Ω( \frac{1}{2} r_{1},2r_{1}))} ≤ c(k, p)r_{1}^{(4/p)−k−1}∥F_{A}∥_{L^{2}(Ω( \frac{1}{4} r_{1},4r_{1}))},
∥∇_{Γ}^{k}F_{A}∥_{L^{p}(Ω( \frac{1}{2} r_{1},2r_{1}))} ≤ c(k, p)r_{1}^{(4/p)−k−2}∥F_{A}∥_{L^{2}(Ω( \frac{1}{4} r_{1},4r_{1}))},

and a C^{∞} bundle isomorphism u : P |_{Ω′} → Ω′ × G taking Γ to the product connection. Finally, if Γ is the product connection on Ω × G, P = Ω × G, and A − Γ is in radial gauge on Ω with respect to the origin, then the preceding Sobolev bounds for A − Γ continue to hold.

Proof. We proceed much as in the proof of Lemma 5.1. Define a rescaled metric g̃ on ℝ^{4} by setting g̃_{μν} = r_{1}^{−2}g_{μν}, so g̃^{μν} = r_{1}^{2}g^{μν} and Ω_{g}(\frac{1}{4}r_{1}, 4r_{1}) = Ωg̃(\frac{1}{4}, 4). Since ∥F_{A}∥_{L^{2}(Ω( \frac{1}{4} ,4),g̃)} < ε

and ε is sufficiently small, Proposition 4.20 provides a C^{∞} flat connection Γ and a C^{∞} bundle isomorphism P |_{Ω′} → Ω′ × G, taking Γ to the product connection, such that (noting that its proof extends from p = 2 to the general case in a straightforward manner)

∥A − Γ∥_{L_{k,Γ}^{p}(Ωg̃( \frac{1}{2} ,2))} ≤ c(k, p)∥F_{A}∥_{L^{2}(Ωg̃( \frac{1}{4} ,4))},
∥F_{A}∥_{L_{k,Γ}^{p}(Ωg̃( \frac{1}{2} ,2))} ≤ c(k, p)∥F_{A}∥_{L^{2}(Ωg̃( \frac{1}{4} ,4))}.

The rescaling behavior of pointwise norms (5.1) implies that

∥∇_{Γ}^{k}(A − Γ)∥_{L^{p}(Ω( \frac{1}{2} r_{1},2r_{1}),g)} = r_{1}^{(4/p)−k−1}∥∇_{Γ}^{k}(A − Γ)|_{L^{2}(Ω( \frac{1}{2} ,2),g̃)},
∥∇_{A}^{k}F_{A}∥_{L^{p}(Ω( \frac{1}{2} r_{1},2r_{1}),g)} = r_{1}^{(4/p)−k−2}∥∇_{A}^{k}F_{A}∥_{L^{2}(Ω( \frac{1}{2} ,2),g̃)}.

Therefore, the desired Sobolev L_{k}^{p} estimates follow by combining the preceding two pairs of estimates.

Lastly, if A − Γ is in radial gauge with respect to the origin, that is x^{μ}A_{μ}(x) = 0 for all x ∈ Ω, then (5.2) yields

A_{ν}(x) = ∫_{r_{1}/4}^{|x|} tθ^{μ}(F_{A})_{μν}(tθ) dt, x ∈ Ω,

where θ = x/|x| ∈ S^{3}. The rest of the proof now mimics that of Lemma 5.2.

5.4. Estimates for cut-off functions. For any N > 1 and λ > 0, Lemma 7.2.10 in [8] provides a clever construction of a cut-off function β on ℝ^{4} such that β(x) = 0 for |x| ≤ N^{−1}\sqrt{λ}, β(x) = 1 for |x| ≥ N\sqrt{λ}, and satisfying the crucial estimate

∥∇β∥_{L^{4}} ≤ c(log N)^{−3/4},

where c is a constant independent of N and λ. We will need the following refinement of this result; for the definition and properties of the L^{♯} and related families of Sobolev norms, see [12, §4].

Lemma 5.8. There is a positive constant c such that the following holds. For any N > 4 and λ > 0, there is a C^{∞} cut-off function β = β_{N,λ} on ℝ^{4} such that

β(x) = \begin{cases}1 & if |x| ≥ \frac{1}{2} \sqrt{λ}, \\ 0 & if |x| ≤ N^{−1}\sqrt{λ}, \end{cases}

and satisfying the following estimates:

(1)  ∥∇β∥_{L^{4}} ≤ c(log N)^{−3/4},
(2)  ∥∇^{2}β∥_{L^{2}} ≤ c(log N)^{−1/2},
(3)  ∥∇β∥_{L^{2♯}} ≤ c(log N)^{−1/2},
(4)  ∥∇β∥_{L^{2}} ≤ c(log N)^{−1}\sqrt{λ},
(5)  ∥∇^{2}β∥_{L^{4/3}} ≤ c(log N)^{−1}\sqrt{λ}.

Proof. We fix a C^{∞} cut-off function κ : ℝ → [0, 1] such that κ(t) = 1 if t ≥ 1 and κ(t) = 0 if t ≤ 0. Now define a C^{∞} cut-off function α = α_{N} : ℝ → [0, 1], depending on the parameter N, by setting

α(t) := κ (\frac{log N + t}{log N − log 2}) .

Therefore,

α(t) = \begin{cases}1 & if t ≥ − log 2, \\ 0 & if t ≤ − log N, \end{cases}

and there is a constant c independent of N such that

|\frac{dα}{dt}| ≤ \frac{c}{log N} and |\frac{d^{2}α}{dt^{2}} | ≤ \frac{c}{(log N)^{2}} .

Now define the cut-off function β = β_{N,λ} : ℝ → [0, 1], depending on the parameters N and λ, by setting

β(x) := α(log |x| − log \sqrt{λ}), x ∈ ℝ^{4},

and observe that β satisfies the following pointwise bounds on ℝ^{4}:

(5.4) |∇β(x)| ≤ \frac{c}{r log N} and |∇^{2}β(x)| ≤ \frac{c}{r^{2} log N} ,

where r = |x|. Consequently, using dV = r^{3} drdθ,

∫_{ℝ^{4}} |∇β|^{4} dV ≤ \frac{c}{(log N)^{4}} ∫_{N^{−1}\sqrt{λ}}^{\frac{1}{2} \sqrt{λ}} \frac{dr}{r} ≤ c\frac{log N − log 2}{(log N)^{4}} ≤ \frac{c}{(log N)^{3}} ,

and so this gives (1). Next, by the pointwise estimate (5.4) we have

∫_{ℝ^{4}} |∇^{2}β|^{2} dV ≤ \frac{c}{(log N)^{2}} ∫_{N^{−1}\sqrt{λ}}^{\frac{1}{2} \sqrt{λ}} \frac{dr}{r} ≤ c\frac{log N − log 2}{(log N)^{2}} ≤ \frac{c}{log N} ,

which gives (2). Lastly, [12, Lemma 5.2] implies that

∥∇β∥_{L^{2♯}} ≤ c(∥∇^{2}β∥_{L^{2}} + ∥∇β∥_{L^{4}}),

and so (3) follows from (1) and (2). Next, the pointwise estimate (5.4) yields

∫_{ℝ^{4}} |∇β|^{2} dV ≤ \frac{c}{(log N)^{2}} ∫_{N^{−1}\sqrt{λ}}^{\frac{1}{2} \sqrt{λ}} rdr ≤ c\frac{λ(1 − N^{−2})}{(log N)^{2}} ≤ \frac{cλ}{(log N)^{2}} ,

and so this gives (4). Finally, the pointwise estimate (5.4) yields

∫_{ℝ^{4}} |∇β|^{4/3} dV ≤ \frac{c}{(log N)^{4/3}} ∫_{N^{−1}\sqrt{λ}}^{\frac{1}{2} \sqrt{λ}} r^{1/3}dr ≤ c\frac{λ^{2/3}(1 − N^{−4/3})}{(log N)^{4/3}} ≤ \frac{cλ^{2/3}}{(log N)^{4/3}} ,

and so this gives (5).

5.5. Estimate for the self-dual curvature of cut-off connections. An important application of our local estimates for anti-self-dual connections is to bound the self-dual curvature of approximately anti-self-dual connections produced by splicing.

Before we proceed to estimate F_{A}^{+}, where A is connection produced by splicing, it is useful to consider a slightly more general situation than that envisaged in the definition of the splicing map γ′ in §3: this will, for example, allow us to apply our self-dual curvature estimates in situations where the gluing maps are iterated and the component connections are generally only anti-self-dual on open sets (where no cutting off occurs). Specifically, we require that the curvatures of the connections to be cut off are L^{2}-small and anti-self-dual near the cut-off points:

Definition 5.9. Let (X, g) be a C^{∞}, closed, oriented, Riemannian manifold, let ℓ < κ be positive integers, and let w ∈ H^{2}(X, ℤ). Let Y ⊂ X be a precompact open subset, let Σ ⊂ Sym^{ℓ}(X) be a smooth stratum, let ε be a positive constant which is less than or equal to the constants of Lemmas 5.1 and 5.5, and let 0 < ϱ ≤ 1 be a constant. We call a connection A suitable for gluing if it has been produced by the splicing construction of §3 (see definition (3.26), for example). The background connection A_{0} and the four-sphere connections A_{i} need not be anti-self-dual but are required to obey the following constraints. Suppose x ∈ Sym^{ℓ}(X) is represented by the m-tuple (x_{1}, …, x_{m}) with multiplicities κ_{i} and ∪_{x∈x}B(x, ϱ) ⊂ Y. Then:

1. The background connection, A_{0}, satisfies • F_{A_{0}}^{+,g} = 0 on Y, and

• sup_{x∈Y} ∥F_{A}∥_{L^{2}(B(x,ϱ))} < ε.

2. Require that ϱ be small enough that ϱ\sqrt{8π^{2}κ_{i}} < ε, for all i ≥ 1. Let δ_{i} be a positive constant, let f_{i} be an orthonormal frame for T X|_{x_{i}}, for each point x_{i} ∈ x, and let g_{i} be the corresponding metric on ℝ^{4} = S^{4} − {s} defined in §3.3.2. Let λ_{i} be the metric-rescaling constant and let g̃_{i} = λ_{i}^{−2}g_{i} be the rescaled metric on ℝ^{4} = S^{4} − {s}. The four-sphere connections, A_{i}, i ≥ 1, satisfy

• F_{A_{i}}^{+,g̃_{i}} = 0 on ℝ^{4} − B(ϱ^{−1}),

• z[A_{i}, g̃_{i}] = 0 and λ[A_{i}, g̃_{i}] = 1.

3. The scales satisfy \sqrt{λ} ≤ ϱ/16, where λ := max_{1≤i≤m} λ_{i}.

Remark 5.10. The requirements of Definition 5.9 are satisfied by connections A produced by the splicing map γ′ described in §3, with domain Gl_{κ}^{w}(X, U_{0}, Σ, λ_{0}). The background connections A_{0} are g-anti-self-dual over X and the four-sphere connections A_{i} are g̃_{i}-anti-self-dual via the deformation described in §9.

We briefly recall some elementary facts concerning metrics, connections, and rescaling. Suppose, to begin, that g is a Euclidean metric on ℝ^{4} and so there are coordinates {x^{μ}} on ℝ^{4} with respect to which g has the standard form, g(x) = δ_{μν}dx^{μ} ⊗dx^{ν} = dx^{μ} ⊗dx^{μ}. Given a positive constant λ, let g̃ = λ^{−2}g denote the rescaled metric, so g̃(x) = λ^{−2}dx^{μ} ⊗ dx^{μ}. With respect to the coordinates y = x/λ, the metric g̃ becomes g̃(y) = dy^{μ}⊗dy^{μ}. Analogous remarks apply to metrics which are not necessarily Euclidean.

Let g be a metric on ℝ^{4} and let λ[A, g] be the scale of a unitary connection A over ℝ^{4} with energy 8π^{2}κ and center of mass at the origin,

λ[A, g]^{2} := \frac{1}{8π^{2}κ} ∫_{ℝ^{4}} dist_{g}(0, ·)^{2}|F_{A}|_{g}^{2} dV_{g}.

With respect to the rescaled metric, g̃ = λ^{−2}g, we see that

λ[A, g̃]^{2} := \frac{1}{8π^{2}κ} ∫_{ℝ^{4}} distg̃(0, ·)^{2}|F_{A}|2 ; g̃ dVg̃ = 1,

since distg̃(0, ·) = λ dist_{g}(0, ·) and |F_{A}|2 ; g̃ dVg̃ = |F_{A}|_{g}^{2} dV_{g}.

Returning to the second part of Definition 5.9, the Chebychev inequality (see Lemma 3.3) implies that

∥F_{A_{i}}∥_{L^{2}(ℝ^{4}−B(0,1/ϱ),g̃_{i})} = ∥F_{A_{i}}∥_{L^{2}(ℝ^{4}−B(0,λ_{i}/ϱ),g_{i})} ≤ ϱ\sqrt{8π^{2}κ_{i}} < ε,

since ϱ\sqrt{8π^{2}κ_{i}} < ε, by definition of ϱ, and as

λ[A_{i}, g_{i}]^{2} = \frac{1}{8π^{2}κ} ∫_{ℝ^{4}} dist_{g_{i}}(0, ·)^{2}|F_{A_{i}}|_{g_{i}}^{2} dV_{g_{i}} = λ_{i}^{2}.

The following proposition gives the key estimate we need:

Proposition 5.11. There is a positive constant c with the following significance. Let A be a spliced connection on 𝔤_{E} which is suitable for gluing, as in Definition 5.9. Then,

∥F_{A}^{+,g}∥_{L^{p}(X,g)} ≤ cελ^{2/p}ϱ^{−2} + c∥F_{A_{0}}^{+,g}∥_{L^{p}(X,g)} + c ∑_{i=1}^{m} ∥F_{A_{i}}^{+,g_{i}}∥_{L^{p}(S^{4},g_{i})}.

If A_{0} is g-anti-self-dual over X, the A_{i} are g_{i}-anti-self-dual over S^{4}, and c′ := cεϱ^{−2},

∥F_{A}^{+,g}∥_{L^{p}(X)} ≤ c′λ^{2/p}. (5.5)

Proof. Recall from definition (3.26) that

(5.6) A := \begin{cases}A_{0} & on X − ∪_{i=1}^{m}B(x_{i}, 4\sqrt{λ_{i}}), \\ Γ + χ_{x_{i},4\sqrt{λ_{i}}}σ_{i}^{∗}A_{0} & on Ω(x_{i}, 2\sqrt{λ_{i}}, 4\sqrt{λ_{i}}), \\ Γ & on Ω(x_{i}, \sqrt{λ_{i}}/2, 2\sqrt{λ_{i}}), \\ Γ + (1 − χ_{x_{i},\sqrt{λ_{i}}/2})(φ_{n} ∘ φ_{i}^{−1})^{∗}τ_{i}^{∗}A_{i} & on Ω(x_{i}, \sqrt{λ_{i}}/4, \sqrt{λ_{i}}/2), \\ (φ_{n} ∘ φ_{i}^{−1})^{∗}A_{i} & on B(x_{i}, \sqrt{λ_{i}}/4), \end{cases}

where the cut-off functions are defined in equation (3.20). Over the region X−∪_{i=1}^{m}B(x_{i}, 4\sqrt{λ_{i}}) we have F_{A} = F_{A_{0}}, while over the annuli Ω(x_{i}, \sqrt{λ_{i}}/2, 2\sqrt{λ_{i}}) we have F_{A} = 0, and over the balls B(x_{i}, \sqrt{λ_{i}}/4) we have F_{A} = F_{A_{i}}. Thus, it remains to consider the two annuli where cutting off occurs.

The hypotheses imply that F_{A_{0}}^{+,g} = 0 on each ball B(x_{i}, ϱ). Over the outer annulus Ω(x_{i}, 2\sqrt{λ_{i}}, 4\sqrt{λ_{i}}) we see that (5.6) yields

F_{A} = χ_{i}F_{A_{0}} + dχ_{i} ∧ σ_{i}^{∗}A_{0} + (χ_{i}^{2} − χ_{i})σ_{i}^{∗}A_{0} ∧ σ_{i}^{∗}A_{0}, (5.7)

where we set χ_{i} = χ_{x_{i},4\sqrt{λ_{i}}} for convenience. According to Lemma 5.1, we have the L^{∞} curvature estimate and by Lemma 5.2 , since the connection one-form σ_{i}^{∗}A_{0} is in radial gauge with respect to the point x_{i}, the L^{p} connection one-form estimate,

∥F_{A_{0}}∥_{L^{∞}(B(x_{i},ϱ/2))} ≤ cϱ^{−2}∥F_{A_{0}}∥_{L^{2}(B(x_{i},ϱ))},
∥σ_{i}^{∗}A_{0}∥_{L^{p}(B(x_{i},4\sqrt{λ_{i}}))} ≤ cλ_{i}^{(2/p)−1/2}∥F_{A_{0}}∥_{L^{2}(B(x_{i},8\sqrt{λ_{i}}))}.

The lemmas are applicable because of the choice of ε in Definition 5.9. Because ∥F_{A_{0}}∥_{L^{2}(B(x_{i},8\sqrt{λ_{i}}))} ≤ cλ_{i}∥F_{A_{0}}∥_{L^{∞}(B(x_{i},8\sqrt{λ_{i}}))}, we can combine the preceding two estimates to get the L^{p} estimate, for any 1 ≤ p ≤ ∞,

∥σ_{i}^{∗}A_{0}∥_{L^{p}(B(x_{i},4\sqrt{λ_{i}}))} ≤ cλ_{i}^{(2/p)+1/2}ϱ^{−2}∥F_{A_{0}}∥_{L^{2}(B(x_{i},ϱ))}, \sqrt{λ_{i}} ≤ ϱ/16. (5.8)

Given 1 ≤ p ≤ ∞, choose 4/3 ≤ q ≤ ∞ by setting 1/p = 1/4 + 1/q. From the expression (5.7) for F_{A} over the annulus Ω(x_{i}, 2\sqrt{λ_{i}}, 4\sqrt{λ_{i}}) and the integral estimates (5.8) for σ_{i}^{∗}A_{0} and that of Lemma 5.8 for dχ_{i}, we obtain an estimate for F_{A}^{+},

(5.9)  ∥F_{A}^{+}∥_{L^{p}(Ω(x_{i},2\sqrt{λ_{i}},4\sqrt{λ_{i}}))} ≤ ∥F_{A_{0}}^{+}∥_{L^{p}} + ∥dχ_{i}∥_{L^{4}}∥σ_{i}^{∗}A_{0}∥_{L^{q}(Ω(x_{i},2\sqrt{λ_{i}},4\sqrt{λ_{i}}))}
+ ∥σ_{i}^{∗}A_{0}∥_{L^{2p}(Ω(x_{i},2\sqrt{λ_{i}},4\sqrt{λ_{i}}))}^{2}
≤ ∥F_{A_{0}}^{+}∥_{L^{p}} + cϱ^{−2}(λ_{i}^{2/p} + λ_{i}^{(2/p)+1})∥F_{A_{0}}∥_{L^{2}(B(x_{i},ϱ))}.

This completes the required estimates for F_{A}^{+} over the outer annuli.

Next we turn to the estimate of F_{A}^{+} over the inner annulus Ω(x_{i}, \sqrt{λ_{i}}/4, \sqrt{λ_{i}}/2). The hypotheses imply that F_{A_{i}}^{+,g_{i}} = 0 on ℝ^{4} − B(λ_{i}/ϱ). This time, our expression (5.6) for the connection A yields

F_{A} = χ_{i}F (τ_{i}^{∗}A_{i}) + dχ_{i} ∧ τ_{i}^{∗}A_{i} + (χ_{i}^{2} − χ_{i})τ_{i}^{∗}A_{i} ∧ τ_{i}^{∗}A_{i}, (5.10)

where we now abbreviate χ_{i} = 1 − χ_{x_{i},\sqrt{λ_{i}}/2} and τ_{i}^{∗}A_{i} = (φ_{n} ∘ φ_{i}^{−1})^{∗}τ_{i}^{∗}A_{i}.

According to Lemma 5.1, we have the L^{∞} curvature estimate and by Lemma 5.5, since the connection one-form τ_{i}^{∗}A_{i} is in radial gauge with respect to the south pole s, the L^{p} connection one-form estimate,

∥F_{A_{i}}∥_{L^{∞}(φ_{s}(B(ϱ/2)),g̃_{i})} ≤ cϱ^{−2}∥F_{A_{i}}∥_{L^{2}(φ_{s}(B(ϱ)),g̃_{i})},
∥τ_{i}^{∗}A_{i}∥_{L^{p}(φ_{n}(ℝ^{4}−B(\sqrt{λ_{i}}/4)),g_{i})} ≤ cλ_{i}^{(2/p)−1/2}∥F_{A_{i}}∥_{L^{2}(φ_{n}(ℝ^{4}−B(x_{i},\sqrt{λ_{i}}/8)),g_{i})}.

Again, the lemmas are applicable because of the choice of ε in Definition 5.9. Since

∥F_{A_{i}}∥_{L^{2}(φ_{s}(B(ϱ)),g̃_{i})} = ∥F_{A_{i}}∥_{L^{2}(φ_{n}(ℝ^{4}−B(1/ϱ)),g̃_{i})} = ∥F_{A_{i}}∥_{L^{2}(φ_{n}(ℝ^{4}−B(λ_{i}/ϱ)),g_{i})},
∥F_{A_{i}}∥_{L^{2}(φ_{s}(B(\sqrt{λ_{i}}/8)),g̃_{i})} ≤ cλ_{i}∥F_{A_{i}}∥_{L^{∞}(φ_{s}(B(\sqrt{λ_{i}}/8)),g̃_{i})},
∥F_{A_{i}}∥_{L^{2}(φ_{n}(ℝ^{4}−B(\sqrt{λ_{i}}/8)),g_{i})} = ∥F_{A_{i}}∥_{L^{2}(φ_{n}(ℝ^{4}−B(1/8\sqrt{λ_{i}})),g̃_{i})} = ∥F_{A_{i}}∥_{L^{2}(φ_{s}(B(8\sqrt{λ_{i}})),g̃_{i})},

we can combine the two preceding sets of estimates to get the L^{p} bound, for any 1 ≤ p ≤ ∞,

∥τ_{i}^{∗}A_{i}∥_{L^{p}(φ_{n}(ℝ^{4}−B(\sqrt{λ_{i}}/4)),g_{i})} ≤ cλ_{i}^{(2/p)+1/2}ϱ^{−2}∥F_{A_{i}}∥_{L^{2}(φ_{s}(B(ϱ)),g̃_{i})}, \sqrt{λ_{i}} ≤ ϱ/16. (5.11)

From the expression (5.10) for F_{A} over the inner annulus Ω(x_{i}, \sqrt{λ_{i}}/4, \sqrt{λ_{i}}/2) and the integral estimates (5.11) for φ_{n}^{∗}τ_{i}^{∗}A_{i} and that of Lemma 5.8 for dχ_{i}, we obtain an estimate for F_{A}^{+},

∥F_{A}^{+}∥_{L^{p}(Ω(x_{i},\sqrt{λ_{i}}/4,\sqrt{λ_{i}}/2))} (5.12)

≤ ∥F_{A_{i}}^{+}∥_{L^{p}(S^{4},g_{i})} + ∥dχ_{i}∥_{L^{4}}∥τ_{i}^{∗}A_{i}∥_{L^{q}(Ω(n,\sqrt{λ_{i}}/4,\sqrt{λ_{i}}/2),g_{i})}
+ ∥τ_{i}^{∗}A_{i}∥_{L^{2p}(Ω(n,\sqrt{λ_{i}}/4,\sqrt{λ_{i}}/2),g_{i})}^{2}
≤ ∥F_{A_{i}}^{+}∥_{L^{p}(S^{4},g_{i})} + cϱ^{−2}(λ_{i}^{2/p} + λ_{i}^{(2/p)+1})∥F_{A_{i}}∥_{L^{2}(Ω(n,λ_{i}/ϱ,∞),g_{i})}.

This completes the required estimate for F_{A}^{+} over the inner annulus.

The required bound for F_{A}^{+} now follows by combining the estimates (5.9) and (5.12) for F_{A}^{+} over the annuli where cutting off occurs, together with the trivial estimates for F_{A}^{+} on the complement of these annuli.

Remark 5.12. A slightly different approach to constructing pointwise estimates for local connection one-forms on annuli (via temporal gauge on cylinders) is described in [8, §7.3.4].

Corollary 5.13. Continue the hypotheses and notation of Proposition 5.11. Then, for any constant 2 < p ≤ ∞,

∥F_{A}^{+,g}∥_{L^{♯,2}(X)} ≤ c′λ^{2/p} and ∥F_{A}^{+,g}∥_{L^{∞}(X)} ≤ c′. (5.13)

where the constant c′ now also depends on the choice of p.

Proof. This follows from the estimate (5.5) of Proposition 5.11, together with the embedding L^{p}(X) ⊂ L^{♯}(X) of [12, Lemma 4.1 (1)].

6. Eigenvalue estimates for Laplacians on self-dual two-forms

We need to analyze the behavior of the small eigenvalues of the Laplacian d_{A}^{+}d_{A}^{+,∗} on L^{2}(X, Λ^{+} ⊗ 𝔤_{E}) as the point [A] in B_{E}(X) converges, in the Uhlenbeck topology, to a point ([A_{0}], x) in B_{E_{0}}(X) × Sym^{ℓ}(X). Our strategy is modelled on the method used by Taubes to describe the small eigenvalues of d_{A}^{+}d_{A}^{+,∗} [54, Lemma 4.8], where A is an approximately anti-self-dual connection constructed by splicing |x| ≤ ℓ one-instantons onto the background product connection Γ. However, the upper and lower bounds for the small eigenvalues we obtain here are more precise and more general than those of [54] or the analogous arguments are used by Mrowka in the proof of Theorem 6.1.0.0.4 [49] in order to establish his ‘main eigenvalue estimate’. Our derivation in [14] of the eigenvalue bounds for the Laplacian d_{A}^{∗}d_{A} on L^{2}(X, 𝔤_{E}) is closely modelled on the arguments given here for the Laplacian d_{A}^{+}d_{A}^{+,∗} on L^{2}(X, Λ^{+} ⊗ 𝔤_{E}).

In the present article, our main application of the eigenvalue bounds of this section will be to deriving an estimate for the partial right inverse P_{A,μ} = d_{A}^{+,∗}(d_{A}^{+}d_{A}^{+,∗})^{−1}Π_{A,μ}^{⊥} for d_{A}^{+}, as given in Proposition 7.3. We have the following analogue of Proposition 4.6 in [54].

Theorem 6.1. Let M be a positive constant and let (X, g) be a closed, oriented, Riemannian four-manifold. Then there are positive constants c = c(g), C = C(g, M), and small enough λ_{0} = λ_{0}(C) such that for all λ ∈ (0, λ_{0}], the following holds. Let E, E_{0} be Hermitian, rank-two vector bundles over X with det E = det E_{0}, c_{2}(E) = c_{2}(E_{0}) + ℓ, for some integer ℓ ≥ 0, and suppose \frac{1}{4}p_{1}(𝔤_{E}) ≤ M. Let [A_{0}] ⊂ B_{E_{0}}(X) be a point such that ∥F_{A_{0}}^{+,g}∥_{L^{∞}(X,g)} < M. Suppose x ∈ Sym^{ℓ}(X) and define a precompact open subset of X \ x by setting U := X − ∪_{x∈x}B̄(x, 4\sqrt{λ}). Let A be a connection representing a point in the open subset of B_{E}(X) defined by the following constraints:

• ∥F_{A}^{+,g}∥_{L^{∞}(X,g)} < M,

• ∥A − A_{0}∥_{L^{2}(U)} < c\sqrt{λ} and ∥A − A_{0}∥_{L_{1,A_{0}}^{2}(U)} < M, (noting that E|_{X\x} ≅ E_{0}|_{X\x}).

Let ν_{2}[A_{0}] be the least positive eigenvalue of the Laplacian d_{A_{0}}^{+}d_{A_{0}}^{+,∗} on L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) and suppose max{ν_{2}[A_{0}], ν_{2}[A_{0}]^{−1}, ∥F_{A_{0}}^{+,g}∥_{L^{∞}(X,g)}} ≤ M. Assume that the kernel of d_{A_{0}}^{+}d_{A_{0}}^{+,∗} has dimension n ≤ M. Let {μ_{i}[A]}_{i≥1} denote the eigenvalues, repeated according to their multiplicity, of the Laplacian d_{A}^{+}d_{A}^{+,∗} on L^{2}(Λ^{+} ⊗ 𝔤_{E}). Then

μ_{i}[A] ≤ \begin{cases}Cλ, & if i = 1, …, n, \\ ν_{2}[A_{0}] + C\sqrt{λ}, & if i = n + 1, \end{cases}
μ_{i}[A] ≥ ν_{2}[A_{0}] − C\sqrt{λ}, if i ≥ n + 1.

Remark 6.2. 1. Recall that by [8, §2.1.4]

κ = −\frac{1}{4}p_{1}(𝔤_{E}) = \frac{1}{8π^{2}} (∥F_{A}^{−}∥_{L^{2}}^{2} − ∥F_{A}^{+}∥_{L^{2}}^{2}) = \frac{1}{8π^{2}} (∥F_{A}∥_{L^{2}}^{2} − 2∥F_{A}^{+}∥_{L^{2}}^{2}),

and therefore,

∥F_{A}∥_{L^{2}}^{2} ≤ 2M^{2} + 8π^{2}κ.

Thus, we have a universal energy bound for all points in B_{E}(X) with ∥F_{A}^{+,g}∥_{L^{∞}(X,g)} < M.

2. If a point [A] in B_{E}(X) lies in an Uhlenbeck neighborhood (see Definition 3.1) of a point ([A_{0}], x) in B_{E_{0}}(X)×Sym^{ℓ}(X), then the hypotheses of Theorem 6.1 are fulfilled.

Corollary 6.3. Let (X, g) be a closed, oriented, Riemannian four-manifold. Suppose [A_{α}] is a sequence of points in B_{E}(X) which converges, in the Uhlenbeck topology, to a point ([A_{0}], x) ∈ B_{E_{0}}(X) × Sym^{ℓ}(X) and suppose the kernel of d_{A_{0}}^{+}d_{A_{0}}^{+,∗} has dimension n. Then the first n eigenvalues (counted with multiplicity) of d_{A_{α}}^{+}d_{A_{α}}^{+,∗} converge to zero, while the (n+1)-st eigenvalue of d_{A_{α}}^{+}d_{A_{α}}^{+,∗} converges to ν_{2}[A_{0}], the least positive eigenvalue of d_{A_{0}}^{+}d_{A_{0}}^{+,∗}.

Remark 6.4. The statements of Theorem 6.1 and Corollary 6.3 extend without difficulty from the case of a single, fixed metric g and connection [A_{0}] to the case of a family U_{0} ⊂ B_{E}(X) × Met(X) satisfying the constraints

• Spec d_{A_{0}}^{+,g}d_{A_{0}}^{+,g,∗} ⊂ [0, t) ∪ (2μ, ∞), for some positive constants t, μ such that t ≤ μ/2 and all [A_{0}, g] ∈ U_{0},

• The number of eigenvalues, counted with multiplicity, of d_{A_{0}}^{+,g}d_{A_{0}}^{+,g,∗} which are less than μ is constant, say n, as [A_{0}, g] varies over U_{0}, and

• ∥F_{A_{0}}^{+,g}∥_{L^{∞}(X,g)} < M. Thus, for example, in Corollary 6.3 we can replace g by a sequence {g_{α}} of smooth Riemannian metrics on X converging in C^{r}, r ≥ 1, to a metric g_{0}. In the statements of Theorem 6.1 and Corollary 6.3, we replace ν_{2}[A_{0}] by μ. Then, in the proofs below, we replace ν_{2}[A_{0}] by μ, Ker d_{A_{0}}^{+,g}d_{A_{0}}^{+,g,∗} by Ran Π_{A_{0},μ} and Π_{A_{0}} by Π_{A_{0},μ}, the projection from L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) onto the span of the eigenvectors of d_{A_{0}}^{+,g}d_{A_{0}}^{+,g,∗} with eigenvalue less than μ. Thus, in the statement of Theorem 6.1, the upper bounds are replaced by C(t + λ) and μ + C(\sqrt{t} + \sqrt{λ}), respectively, while the lower bound is replaced by μ − C(\sqrt{t} + \sqrt{λ}), for all t < t_{0}(C) and λ < λ_{0}(C). Similar modifications apply to Corollary 6.3, noting that λ is replaced by a sequence {λ_{α}} converging to zero.

The proof of Theorem 6.1 takes up the remainder of this section. While the argument is lengthy, the basic strategy is reasonably straightforward:

• Find upper bounds for the first n + 1 eigenvalues of d_{A}^{+}d_{A}^{+,∗};

• Construct an approximately L^{2}-orthonormal basis for Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} by cutting off the eigenvectors corresponding to the first n eigenvalues of d_{A}^{+}d_{A}^{+,∗};

• Compute a lower bound for the (n + 1)-st eigenvalue of d_{A}^{+}d_{A}^{+,∗}.

We begin in the next subsection by disposing of a few technical lemmas.

6.1. Technical preliminaries. We begin with some elementary lemmas before proceeding to the different stages of the proof of Theorem 6.1. The following Kato-Sobolev inequality is well-known; see Lemma 4.6 in [52] for the case p = 2.

Lemma 6.5. Let X be a C^{∞}, closed, oriented, Riemannian, four-manifold and let 1 ≤ p < 4. Then there is a positive constant c with the following significance. Let A be an orthogonal L_{1}^{p} connection on a Riemannian vector bundle V over X. Then, for any a ∈ L_{1}^{p}(V), we have

∥a∥_{L^{q}(X)} ≤ c∥a∥_{L_{1,A}^{p}(X)},

when 1/p = 1/q + 1/4.

Proof. The estimate follows from the pointwise Kato inequality |∇|a|| ≤ |∇_{A}a| (see [24, Equation (6.20)] and the Sobolev inequality

∥f∥_{L^{q}} ≤ c(∥∇f∥_{L^{q}} + ∥f∥_{p}),

for any f ∈ L_{1}^{p}(X) [2, Theorem 5.4].

We shall need the Bochner-Weitzenböck formulas,

(6.1)  d_{A}d_{A}^{∗} + 2d_{A}^{+,∗}d_{A}^{+} = ∇_{A}^{∗}∇_{A} + {Ric, ·} − 2{F_{A}^{−}, ·},
(6.2)  2d_{A}^{+}d_{A}^{+,∗} = ∇_{A}^{∗}∇_{A} − 2\{W^{+}, ·\} + \frac{R}{3} + \{F_{A}^{+}, ·\},

for the Laplacians on Γ(Λ^{1} ⊗𝔤_{E}) and Γ(Λ^{+} ⊗𝔤_{E}), respectively; see [24, p. 94]. We give L_{1,A}^{2} estimates for v in terms of the L^{2} norm of d_{A}^{+,∗}; similar estimates are given in [52, Lemma 5.2] and in [54, Appendix A].

Lemma 6.6. Let X be a C^{∞}, closed, oriented, Riemannian four-manifold. Then there is a positive constant c with the following significance. Let A be an L_{4}^{2} orthogonal connection on an SO(3) bundle 𝔤_{E} over X. Then, for all v ∈ L_{2}^{4/3}(Λ^{+} ⊗ 𝔤_{E}),

(1)  ∥v∥_{L_{1,A}^{2}(X)} ≤ c(1 + ∥F_{A}^{+}∥_{L^{∞}(X)})^{1/2}(∥d_{A}^{+,∗}v∥_{L^{2}(X)} + ∥v∥_{L^{2}(X)}),
(2)  ∥d_{A}^{+,∗}v∥_{L^{2}(X)} ≤ (1 + ∥F_{A}^{+}∥_{L^{∞}(X)})^{1/2}∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{4/3}(X)}.

Proof. The Bochner-Weitzenbock formula (6.2) and integration by parts yields

∥∇_{A}v∥_{L^{2}}^{2} = (∇_{A}^{∗}∇_{A}v, v)_{L^{2}}

= 2(d_{A}^{+}d_{A}^{+,∗}v, v)_{L^{2}} + (\{W^{+}, v\}, v)_{L^{2}} − \frac{1}{3}(Rv, v)_{L^{2}} − (\{F_{A}^{+}, v\}, v)_{L^{2}}

≤ 2∥d_{A}^{+,∗}v∥_{L^{2}}^{2} + c(1 + ∥F_{A}^{+}∥_{L^{∞}})∥v∥_{L^{2}}^{2},

which gives the first assertion. The second assertion follows by integrating by parts, Hölder’s inequality, and Kato’s inequality (Lemma 6.5),

∥d_{A}^{+,∗}v∥_{L^{2}}^{2} = (d_{A}^{+}d_{A}^{+,∗}v, v)_{L^{2}} ≤ ∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{4/3}}∥v∥_{L^{4}} ≤ c∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{4/3}}∥v∥_{L_{1,A}^{2}},

together with the first inequality.

The following consequence of [12, Lemma 5.9] is the principal tool we use to obtain the required eigenvector estimates.

Lemma 6.7. Let X be a C^{∞}, closed, oriented, Riemannian four-manifold. Then there is a constant c with the following significance. Let A be an L_{4}^{2} connection on 𝔤_{E} and let η = η(A) ∈ L_{3}^{2}(Λ^{+} ⊗ 𝔤_{E}) be an eigenvector of d_{A}^{+}d_{A}^{+,∗} with eigenvalue μ[A]. Then

∥η∥_{L^{∞}(X)} + ∥η∥_{L_{2,A}^{2}(X)} ≤ C[A]∥η∥_{L^{2}(X)},

where C[A] := C([A], μ[A]) with μ = μ[A] in

C([A], μ) := c(1 + ∥F_{A}∥_{L^{2}(X)})^{2}(1 + ∥F_{A}^{+}∥_{L^{∞}(X)})^{5/2}(1 + μ)(1 + \sqrt{μ}).

More generally, if η ∈ L^{2}(Λ^{+} ⊗ 𝔤_{E}) lies in the span of the eigenvectors of d_{A}^{+}d_{A}^{+,∗} with eigenvalues less than or equal to a positive constant μ, then

∥η∥_{L^{∞}(X)} + ∥η∥_{L_{2,A}^{2}(X)} ≤ C([A], μ)∥η∥_{L^{2}(X)}.

Proof. By [12, Lemma 5.9] (or Lemma 7.1 here),

∥η∥_{L^{∞}} + ∥η∥_{L_{2,A}^{2}(X)}
≤ c(1 + ∥F_{A}∥_{L^{2}})^{2}(1 + ∥F_{A}^{+}∥_{L^{∞}})^{2}(∥η∥_{L^{2}} + ∥d_{A}^{+}d_{A}^{+,∗}η∥_{L^{♯}})
= c(1 + ∥F_{A}∥_{L^{2}})^{2}(1 + ∥F_{A}^{+}∥_{L^{∞}})^{2}(1 + μ[A])∥η∥_{L^{♯,2}},

using d_{A}^{+}d_{A}^{+,∗}η = μ[A]η in the second line. On the other hand,

∥η∥_{L^{♯,2}} ≤ c∥η∥_{L^{4}} (by [12, Lemma 4.1])
≤ c∥η∥_{L_{1,A}^{2}} (by Lemma 6.5)
≤ c(1 + ∥F_{A}^{+}∥_{L^{∞}})^{1/2}(∥η∥_{L^{2}} + ∥d_{A}^{+,∗}η∥_{L^{2}}) (by Lemma 6.6)
= c(1 + ∥F_{A}^{+}∥_{L^{∞}})^{1/2}(1 + \sqrt{μ[A]})∥η∥_{L^{2}},

using ∥d_{A}^{+,∗}η∥_{L^{2}} = \sqrt{μ[A]}∥η∥_{L^{2}} to obtain the final equality. Combining the preceding estimates completes the proof of the first inequality.

Next suppose that η ∈ L^{2}(Λ^{+} ⊗𝔤_{E}) lies in the span of the n L^{2}-orthonormal eigenvectors η_{i}(A) of d_{A}^{+}d_{A}^{+,∗} with eigenvalues μ_{i}[A] ≤ μ. Since η = ∑_{i=1}^{n}(η, η_{i})_{L^{2}}η_{i} and denoting C′ = (1 + ∥F_{A}∥_{L^{2}})^{2}(1 + ∥F_{A}^{+}∥_{L^{∞}})^{5/2}, we have

∥η∥_{L^{∞}} ≤ ∑_{i=1}^{n} |(η, η_{i})_{L^{2}}| · ∥η_{i}∥_{L^{∞}}

≤ C′ ∑_{i=1}^{n} |(η, η_{i})_{L^{2}}| · (1 + μ_{i})(1 + \sqrt{μ_{i}})∥η_{i}∥_{L^{2}} (by the special case),

≤ C′(1 + μ)(1 + \sqrt{μ}) (∑_{i=1}^{n}(η, η_{i})_{L^{2}}^{2})^{1/2}

= C′(1 + μ)(1 + \sqrt{μ})∥η∥_{L^{2}},

where we use the inequality (∑_{i=1}^{n} a_{i})^{2} ≤ 2 ∑_{i=1}^{n} a_{i}^{2}, for a_{i} ∈ ℝ, in the second last line. This completes the proof.

The other basic tool we shall need is Courant’s min-max principle for eigenvalues (see, for example, [4, pp. 16-17]).

Lemma 6.8. Let T be a non-negative, self-adjoint operator on a Hilbert space, ℌ, such that (id + T)^{−1} is compact. Let 0 ≤ μ_{1} ≤ μ_{2} ≤ ⋯ be the eigenvalues of T, repeated according to their multiplicity. Then, for any subspace V_{n} ⊂ ℌ with dim V_{n} = n,

μ_{n+1} ≥ inf_{η∈V_{n}^{⊥}} \frac{(η, T η)}{∥η∥^{2}},

where V_{n}^{⊥} is the orthogonal complement of V_{n} in ℌ. If V_{n} is the span of the eigenvectors of T corresponding to the eigenvalues λ_{1}, …, λ_{n}, then equality holds above.

6.2. Good sequences of cut-off functions. In many of our arguments involving Uhlenbeck continuity of eigenvalues, eigenvectors, gluing maps, gluing-map differentials and so on, we shall need suitable sequences of cut-off functions to compare connections, and objects depending on them, which live on different bundles but which become isomorphic when restricted to an open subset X \ x, for some x ∈ Sym^{•}(X). Since the construction of such sequences will arise many times, it is useful to extract the essential properties we shall need once and for all in the following definition-lemma:

Lemma 6.9. Let X be a smooth Riemannian four-manifold and let x ∈ Sym^{•}(X). Then there is a sequence of cut-off functions {χ_{n}}_{n=1}^{∞} such that the following hold:

• ∪_{n=1}^{∞}U_{n} = X \x, where U_{n} ⋐ X \x is the open subset given by the interior of supp χ_{n},

• χ_{m} ≤ χ_{n} for all m ≤ n, so U_{m} ⊂ U_{n} and supp χ_{m} ⊂ supp χ_{n},

• supp χ_{n} ⋐ X \ x,

• lim_{n→∞}(∥∇χ_{n}∥_{L_{1}^{2}} + ∥∇^{2}χ_{n}∥_{L^{♯}}) = 0,

We call any sequence of cut-off functions satisfying the above properties an x-good sequence.

Proof. We may assume without loss that the multiset x ∈ Sym^{•}(X) is represented by a single point x_{0} ∈ X. Recall from Lemma 5.8 that we can define a cut-off function β_{N,λ} : X → ℝ with the property that

β_{N,λ}(x) = \begin{cases}1 & if dist(x, x_{0}) ≥ \frac{1}{2} \sqrt{λ}, \\ 0 & if dist(x, x_{0}) ≤ N^{−1}\sqrt{λ}, \end{cases}

and satisfying the estimates

∥∇β_{N,λ}∥_{L^{4}} ≤ c(log N)^{−3/4},
∥∇^{2}β_{N,λ}∥_{L^{2}} ≤ c(log N)^{−1/2},
∥∇β_{N,λ}∥_{L^{2♯}} ≤ c(log N)^{−1/2},
∥∇β_{N,λ}∥_{L^{2}} ≤ c(log N)^{−1}\sqrt{λ}.

If we choose N = n and λ = n^{−4}, for example, so χ_{n} := β_{n,n^{−4}} and U_{n} = X \ B̄(x_{0}, \frac{1}{2}n^{−2}), then it is immediate that the sequences {χ_{n}}_{n=1}^{∞} and {U_{n}}_{n=1}^{∞} have the desired properties.

6.3. An upper bound for the small eigenvalues. We turn to the proof of Theorem 6.1. We define an “x-good” cut-off function χ_{0} on X (see Lemma 6.9) by setting χ_{0} = χ_{x_{i},8\sqrt{λ_{i}}} (see equation (3.20) for its definition) on each ball B(x_{i}, 8\sqrt{λ_{i}}), so χ_{0} is equal to zero on the balls B(x_{i}, 4\sqrt{λ_{i}}), and setting χ_{0} = 1 on the complement of these balls, X−∪_{i=1}^{m}B(x_{i}, 8\sqrt{λ_{i}}). We take

U := X − ⋃_{i=1}^{m} B(x_{i}, 4\sqrt{λ_{i}}),

so U ⋐ X \ x and U is the interior of supp χ_{0}.

Though it suffices to consider sequences of x-good cut-off functions only satisfying the broad properties of Lemma 6.9, we assume for simplicity that χ_{0} is defined as in the preceding paragraph and observe that supp χ_{0} ⋐ X \ x. We may assume without loss that E|_{X\x} = E_{0}|_{X\x} and thus, over X \ x, we may write

a := A − A_{0} ∈ L_{k,loc}^{2}(X \ x, Λ^{1} ⊗ 𝔤_{E_{0}}), (6.3)

where the section a may also be viewed as an element of L_{k,loc}^{2}(X \ x, Λ^{1} ⊗ 𝔤_{E}). Our initial task is to find an upper bound for the first n + 1 eigenvalues of d_{A}^{+}d_{A}^{+,∗}.

Lemma 6.10. Continue the hypotheses and notation of Proposition 6.1. Let ν_{2}[A_{0}] be the least positive eigenvalue of d_{A_{0}}^{+}d_{A_{0}}^{+,∗} and let μ_{1}[A] ≤ ⋯ ≤ μ_{n+1}[A] be the first n + 1 eigenvalues of d_{A}^{+}d_{A}^{+,∗}. Then there are positive constants C = C(∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}, ν_{2}[A_{0}]) and λ_{0} = λ_{0}(C) such that for all λ ∈ (0, λ_{0}],

μ_{i}[A] ≤ \begin{cases}Cλ, & if i = 1, …, n, \\ ν_{2}[A_{0}] + C\sqrt{λ}, & if i = n + 1. \end{cases} (6.4)

Proof. Let {ξ_{i}(A_{0})}_{i=1}^{n} ⊂ L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) be an L^{2}-orthonormal basis for Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} and let ξ_{n+1}[A_{0}] ∈ (Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗})^{⊥} be an L^{2} unit eigenvector corresponding to the least positive eigenvalue ν_{2}[A_{0}] of d_{A_{0}}^{+}d_{A_{0}}^{+,∗}. We begin by constructing a set of approximate eigenvectors, {ξ′_{i}}_{i=1}^{n+1}, for the first n+1 eigenvalues of d_{A}^{+}d_{A}^{+,∗} by setting ξ′_{i} := χ_{0}ξ_{i} for each i ∈ {1, …, n+ 1}.

Claim 6.11. Continue the above notation. Then there are positive constants C = C(∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}, ν_{2}[A_{0}]) and λ_{0} = λ_{0}(C) such that for all λ ∈ (0, λ_{0}],

(1)  ∥d_{A}^{+,∗}ξ′_{i}∥_{L^{2}(X)} ≤ \begin{cases}C\sqrt{λ}, & if 1 ≤ i ≤ n, \\ \sqrt{ν_{2}[A_{0}]} + C\sqrt{λ}, & if i = n + 1, \end{cases}
(2)  |(ξ′_{i}, ξ′_{j})_{L^{2}(X)} − δ_{ij}| ≤ Cλ^{2}, 1 ≤ i, j ≤ n + 1.

Proof. To verify the first assertion in Claim 6.11, note that A = A_{0} + a on X \ x and d_{A}^{+,∗} = ∗d_{A} on Γ(Λ^{+} ⊗ 𝔤_{E}), so

\begin{aligned}d_{A}^{+,∗}ξ′_{i} & = ∗d_{A}ξ′_{i} = ∗d_{A_{0}}(χ_{0}ξ_{i}) + ∗(a ∧ χ_{0}ξ_{i}) \\ & = ∗(dχ_{0} ∧ ξ_{i}) + χ_{0}d_{A_{0}}^{+,∗}ξ_{i} + ∗(χ_{0}a ∧ ξ_{i}), \end{aligned} (6.5)

while

∥d_{A_{0}}^{+,∗}ξ_{i}∥_{L^{2}} = \begin{cases}0, & if 1 ≤ i ≤ n, \\ \sqrt{ν_{2}[A_{0}]}∥ξ_{i}∥_{L^{2}}, & if i = n + 1. \end{cases}

By Lemma 5.8 (yielding ∥dχ_{0}∥_{L^{2}} ≤ c\sqrt{λ}), Lemma 6.7, the preceding eigenvalue estimate, and equation (6.5),

∥d_{A}^{+,∗}ξ′_{i}∥_{L^{2}} ≤ (∥dχ_{0}∥_{L^{2}} + ∥χ_{0}a∥_{L^{2}})∥ξ_{i}∥_{L^{∞}} + ∥d_{A_{0}}^{+,∗}ξ_{i}∥_{L^{2}}
≤ c\sqrt{λ}∥ξ_{i}∥_{L^{∞}} + ∥d_{A_{0}}^{+,∗}ξ_{i}∥_{L^{2}} (by Lemma 5.8 and hypothesis)
≤ C\sqrt{λ}∥ξ_{i}∥_{L^{2}} + ∥d_{A_{0}}^{+,∗}ξ_{i}∥_{L^{2}} (by Lemma 6.7)
= C\sqrt{λ}∥ξ_{i}∥_{L^{2}} + \sqrt{ν_{2}[A_{0}]}∥ξ_{i}∥_{L^{2}}, 1 ≤ i ≤ n + 1,

and so the first assertion in Claim 6.11 follows. Using ξ′_{i} = χ_{0}ξ = ξ_{i} + (χ_{0} − 1)ξ_{i},

(ξ′_{i}, ξ′_{j})_{L^{2}} = (ξ_{i}, ξ_{j})_{L^{2}} + 2((χ_{0} − 1)ξ_{i}, ξ_{j})_{L^{2}} + ((χ_{0} − 1)ξ_{i}, (χ_{0} − 1)ξ_{j})_{L^{2}}.

But (ξ_{i}, ξ_{j})_{L^{2}} = δ_{ij}, while the second two terms on the right above are bounded by

|2((χ_{0} − 1)ξ_{i}, ξ_{j})_{L^{2}} + ((χ_{0} − 1)ξ_{i}, (χ_{0} − 1)ξ_{j})_{L^{2}}|
≤ cλ^{2}∥ξ_{i}∥_{L^{∞}}∥ξ_{j}∥_{L^{∞}} ≤ Cλ^{2},

courtesy of Lemma 6.7. This gives the second assertion in Claim 6.11.

Now ∥ξ′_{i}∥_{L^{2}} ≥ 1 − Cλ by the second assertion of Claim 6.11 and so its first assertion yields

∥d_{A}^{+,∗}ξ′_{i}∥_{L^{2}} ≤ C\sqrt{λ}(1 − Cλ)^{−1}∥ξ′_{i}∥_{L^{2}}
≤ C\sqrt{λ}∥ξ′_{i}∥_{L^{2}}, 1 ≤ i ≤ n (choosing λ < C/2),
∥d_{A}^{+,∗}ξ′_{n+1}∥_{L^{2}} ≤ (\sqrt{ν_{2}[A_{0}]} + C\sqrt{λ}) ∥ξ′_{n+1}∥_{L^{2}}.

Let V_{i} := [ξ′_{1}, …, ξ′_{i}] denote the span of the vectors ξ′_{1}, …, ξ′_{i} in L^{2}(Λ^{+} ⊗ 𝔤_{E}) for i = 1, …, n + 1. By applying Gram-Schmidt orthonormalization to the basis {ξ′_{1}, …, ξ′_{i}} for V_{i}, the preceding estimates imply that

∥d_{A}^{+,∗}η∥_{L^{2}} ≤ \begin{cases}C\sqrt{λ}∥η∥_{L^{2}}, & if η ∈ V_{n}, \\ (\sqrt{ν_{2}[A_{0}]} + C\sqrt{λ}) ∥η∥_{L^{2}}, & if η ∈ V_{n+1}. \end{cases} (6.6)

Recall from the statement of Theorem 6.1 that μ_{1}[A] ≤ ⋯ ≤ μ_{n+1}[A] are the first n + 1 eigenvalues of d_{A}^{+}d_{A}^{+,∗}; let η_{1}(A), …, η_{n+1}(A) be the corresponding L^{2}-orthonormal eigenvectors. Let [η_{1}, …, η_{i}] denote the span of the eigenvectors η_{1}, …, η_{i} in L^{2}(Λ^{+} ⊗𝔤_{E}) and let [η_{1}, …, η_{i}]^{⊥} be its L^{2}-orthogonal complement in L^{2}(Λ^{+} ⊗𝔤_{E}). Note that dim([η_{1}, …, η_{i}]^{⊥} ∩ V_{n}) ≥ n − i. By the estimates (6.6) we have

μ_{i}[A] = inf_{η∈[η_{1},...,η_{i−1}]^{⊥}} \frac{∥d_{A}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}} ≤ inf_{η∈[η_{1},...,η_{i−1}]^{⊥}∩V_{n}} \frac{∥d_{A}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}} ≤ Cλ,

for i = 1, …, n, which gives the first inequality in the stated eigenvalue bounds (6.4). Therefore, d_{A}^{+}d_{A}^{+,∗} has at least n eigenvalues which are less than or equal to Cλ. Similarly, the estimates (6.6) yield

μ_{n+1}[A] = inf_{η∈[η_{1},...,η_{n}]^{⊥}} \frac{∥d_{A}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}} ≤ inf_{η∈[η_{1},...,η_{n}]^{⊥}∩V_{n+1}} \frac{∥d_{A}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}} ≤ (\sqrt{ν_{2}[A_{0}]} + C\sqrt{λ})^{2} ,

completing the proof of the second inequality in the eigenvalue bounds (6.4).

6.4. An approximately orthonormal basis of eigenvectors. We next construct an approximately L^{2}-orthonormal basis for Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} using the eigenvectors corresponding to the first n eigenvalues of d_{A}^{+}d_{A}^{+,∗}. Let Π_{A_{0}} be the L^{2}-orthogonal projection from L^{2}(Λ^{+}⊗𝔤_{E_{0}}) onto Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} and let Π_{A_{0}}^{⊥} := 1−Π_{A_{0}} be the L^{2}-orthogonal projection from L^{2}(Λ^{+}⊗𝔤_{E_{0}}) onto (Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗})^{⊥} = Ran d_{A_{0}}^{+}d_{A_{0}}^{+,∗}.

Recall from the proof of Lemma 6.10 that {η_{1}[A], …, η_{n+1}[A]} were defined to be the L^{2}- orthonormal eigenvectors associated with the eigenvalues μ_{1}[A] ≤ ⋯ ≤ μ_{n+1}[A] of d_{A}^{+}d_{A}^{+,∗} and construct an approximately L^{2}-orthonormal subset {η′_{i}}_{i=1}^{n+1} ⊂ L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) by setting η′_{i} := χ_{0}η_{i} for i = 1, …, n + 1.

Lemma 6.12. Continue the hypotheses and notation of Theorem 6.1. Then there are positive constants C = C(∥F_{A}∥_{L^{2}}, ∥F_{A}^{+}∥_{L^{∞}}, ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}, ν_{2}[A_{0}]) and λ_{0} = λ_{0}(C) such that for all λ ∈ (0, λ_{0}],

∥Π_{A_{0}}^{⊥}η′_{i}∥_{L^{2}} ≤ C\sqrt{λ}, (1)

|(Π_{A_{0}}η′_{i}, Π_{A_{0}}η′_{j})_{L^{2}(X)} − δ_{ij}| ≤ Cλ, 1 ≤ i, j ≤ n, (2)

∥Π_{A_{0}}η′_{n+1}∥_{L^{2}(X)} ≤ C\sqrt{λ} and ∥Π_{A_{0}}^{⊥}η′_{n+1}∥_{L^{2}(X)} ≥ 1 − C\sqrt{λ}. (3)

Proof. The following estimates are the key to the proof of Lemma 6.12.

Claim 6.13. Continue the above notation. Then there are positive constants C = C(∥F_{A}∥_{L^{2}}, ∥F_{A}^{+}∥_{L^{∞}}, ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}, ν_{2}[A_{0}]) and λ_{0} = λ_{0}(C) such that for all λ ∈ (0, λ_{0}],

(1)  ∥d_{A_{0}}^{+,∗}η′_{i}∥_{L^{2}(X)} ≤ C\sqrt{λ}, 1 ≤ i ≤ n,
(2)  ∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}η′_{i}∥_{L^{4/3}(X)} ≤ C\sqrt{λ}, 1 ≤ i ≤ n,
(3)  |(η′_{i}, η′_{j})_{L^{2}(X)} − δ_{ij}| ≤ Cλ^{2}, 1 ≤ i, j ≤ n + 1.

Proof. To verify the first assertion in Claim 6.13, note that d_{A_{0}}^{+,∗} = ∗d_{A_{0}} on Γ(Λ^{+} ⊗ 𝔤_{E_{0}}), and so

\begin{aligned}d_{A_{0}}^{+,∗}η′_{i} & = d_{A}^{+,∗}(χ_{0}η_{i}) − ∗(a ∧ χ_{0}η_{i}) \\ & = ∗(dχ_{0} ∧ η_{i}) + χ_{0}d_{A}^{+,∗}η_{i} − ∗(χ_{0}a ∧ η_{i}), i = 1, …, n + 1, \end{aligned} (6.7)

where we use the fact that A = A_{0} +a on X \x. Thus, by the estimates in (6.4) (see Lemma 6.10) and Lemmas 5.8 and 6.7, our hypotheses, and the identity (6.7),

∥d_{A_{0}}^{+,∗}η′_{i}∥_{L^{2}} ≤ (∥dχ_{0}∥_{L^{2}} + ∥χ_{0}a∥_{L^{2}})∥η_{i}∥_{L^{∞}} + ∥d_{A}^{+,∗}η_{i}∥_{L^{2}}
≤ c(\sqrt{λ}∥dχ_{0}∥_{L^{4}} + ∥χ_{0}a∥_{L^{2}})∥η_{i}∥_{L^{∞}} + \sqrt{μ_{i}[A]}
≤ C\sqrt{λ}, 1 ≤ i ≤ n,

using ∥d_{A}^{+,∗}η_{i}∥_{L^{2}} = \sqrt{μ_{i}[A]}∥η_{i}∥_{L^{2}}; this proves the first assertion in Claim 6.13. Next, observe that schematically (and again using the fact that A = A_{0} + a on X \ x)

d_{A_{0}}^{+}d_{A_{0}}^{+,∗}η′_{i} = d_{A−a}^{+}(∗(dχ_{0} ∧ η_{i}) + χ_{0}d_{A}^{+,∗}η_{i} − ∗(χ_{0}a ∧ η_{i}))
= (∇^{2}χ_{0} + χ_{0}∇_{A_{0}}a + χ_{0}a ⊗ a) ⊗ η_{i}
+ ∇χ_{0} ⊗ (a ⊗ η_{i} + ∇_{A}η_{i}) + χ_{0}d_{A}^{+}d_{A}^{+,∗}η_{i},

and so, using Lemmas 5.8, 6.5, 6.7, and 6.10, we get

∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}η′_{i}∥_{L^{4/3}} ≤ (∥∇^{2}χ_{0}∥_{L^{4/3}} + ∥χ_{0}∇_{A_{0}}a∥_{L^{4/3}}
+∥∇χ_{0}∥_{L^{2}}∥a∥_{L^{4}(supp χ_{0})} + ∥\sqrt{χ_{0}}a∥_{L^{2}}∥\sqrt{χ_{0}}a∥_{L^{4}}) ∥η_{i}∥_{L^{∞}}
+ c∥∇χ_{0}∥_{L^{2}}∥∇_{A}η_{i}∥_{L^{4}} + ∥d_{A}^{+}d_{A}^{+,∗}η_{i}∥_{L^{2}}
≤ C\sqrt{λ}∥η_{i}∥_{L^{∞}∩L_{1,A}^{4}} + μ_{i}[A]
≤ C\sqrt{λ}, 1 ≤ i ≤ n.

Hence, the second assertion in Claim 6.13 follows from Lemma 5.8 and our hypotheses. Using η′_{i} = χ_{0}η_{i} = η_{i} + (χ_{0} − 1)η_{i},

(η′_{i}, η′_{j})_{L^{2}} = (η_{i}, η_{j})_{L^{2}} + 2((χ_{0} − 1)η_{i}, η_{j})_{L^{2}} + ((χ_{0} − 1)η_{i}, (χ_{0} − 1)η_{j})_{L^{2}}.

Now (η_{i}, η_{j})_{L^{2}} = δ_{ij}, while the second two terms on the right above are bounded by

|2((χ_{0} − 1)η_{i}, η_{j})_{L^{2}} + ((χ_{0} − 1)η_{i}, (χ_{0} − 1)η_{j})_{L^{2}}|
≤ cλ^{2}∥η_{i}∥_{L^{∞}}∥η_{j}∥_{L^{∞}} ≤ Cλ^{2},

and this gives the third assertion in Claim 6.13.

58 PAUL M. N. FEEHAN AND THOMAS G. LENESS December 14, 1998

By Lemmas 6.5 and 6.6, integration by parts, and Claim 6.13 we have, for any ξ ∈ (Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗})^{⊥},

(η′_{i}, d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ξ)_{L^{2}} = (d_{A_{0}}^{+}d_{A_{0}}^{+,∗}η′_{i}, ξ)_{L^{2}} ≤ ∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}η′_{i}∥_{L^{4/3}}∥ξ∥_{L^{4}}
≤ C\sqrt{λ} (∥ξ∥_{L^{2}} + ∥d_{A_{0}}^{+,∗}ξ∥_{L^{2}})
≤ C\sqrt{λ} (∥ξ∥_{L^{2}} + ∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ξ∥_{L^{2}})
≤ C(ν_{0}[A_{0}]^{−1} + 1)\sqrt{λ}∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ξ∥_{L^{2}},

and therefore we obtain (η′_{i}, d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ξ)_{L^{2}} ≤ C\sqrt{λ}∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ξ∥_{L^{2}}, ξ ∈ (Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗})^{⊥}, i = 1, …, n. (6.8) Since Ran Π_{A_{0}}^{⊥} = Ran d_{A_{0}}^{+}d_{A_{0}}^{+,∗}, we may choose ρ_{i} ∈ L_{2}^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) such that Π_{A_{0}}^{⊥}η′_{i} = d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ρ_{i}. If θ_{i} denotes the angle between η′_{i} and Π_{A_{0}}^{⊥}η′_{i} in L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}), then

∥Π_{A_{0}}^{⊥}η′_{i}∥_{L^{2}} = ∥η′_{i}∥_{L^{2}} cos θ_{i} = \frac{(η′_{i}, d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ρ_{i})_{L^{2}}}{∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ρ_{i}∥_{L^{2}}} .

Together with equation (6.8), this yields for i = 1, …, n, ∥Π_{A_{0}}^{⊥}η′_{i}∥_{L^{2}} ≤ C\sqrt{λ}. (6.9) By Claim 6.13 we have, for 1 ≤ i, j ≤ n,

|(Π_{A_{0}}η′_{i}, Π_{A_{0}}η′_{j})_{L^{2}} − δ_{ij}| ≤ |(Π_{A_{0}}η′_{i}, Π_{A_{0}}η′_{j})_{L^{2}} − (η′_{i}, η′_{j})_{L^{2}} | + |(η′_{i}, η′_{j})_{L^{2}} − δ_{ij}|
≤ |(Π_{A_{0}}^{⊥}η′_{i}, Π_{A_{0}}^{⊥}η′_{j})_{L^{2}} | + Cλ^{2}
≤ ∥Π_{A_{0}}^{⊥}η′_{i}∥_{L^{2}}∥Π_{A_{0}}^{⊥}η′_{j}∥_{L^{2}} + Cλ^{2}
≤ C(λ + λ^{2}) (from equation (6.9)),

and so |(Π_{A_{0}}η′_{i}, Π_{A_{0}}η′_{j})_{L^{2}} − δ_{ij}| ≤ Cλ, 1 ≤ i, j ≤ n. (6.10) Thus, for small enough λ, the vectors {Π_{A_{0}}η′_{i}}_{i=1}^{n} form an approximately L^{2}-orthonormal basis for Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗}.

Recall from the proof of Lemma 6.10 that {ξ_{i}(A_{0})}_{i=1}^{n} ⊂ L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) denotes an ordered, L^{2}-orthonormal basis for Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗}. We can therefore assume without loss of generality that {ξ_{i}}_{i=1}^{n} is the basis for Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} obtained by applying Gram-Schmidt orthonormalization to the basis {Π_{A_{0}}η′_{i}}_{i=1}^{n}:

ξ̃_{ℓ} = Π_{A_{0}}η′_{ℓ} − ∑_{i=1}^{ℓ−1}(Π_{A_{0}}η′_{ℓ}, ξ_{i})_{L^{2}}ξ_{i} and ξ_{ℓ} = \frac{ξ̃_{ℓ}}{∥ξ̃_{ℓ}∥_{L^{2}}} , 1 ≤ ℓ ≤ n. (6.11)

By Claim 6.13 and the orthogonal-complement estimates (6.9) we have

(6.12)  |(η′_{n+1}, Π_{A_{0}}η′_{i})_{L^{2}}| = |(η′_{n+1}, η′_{i})_{L^{2}} − (η′_{n+1}, Π_{A_{0}}^{⊥}η′_{i})_{L^{2}}|
≤ Cλ^{2} + C(1 + λ)\sqrt{λ} ≤ C\sqrt{λ}.

Consequently, by the Gram-Schmidt equations (6.11) together with the inner-product estimates (6.12), and the fact that

Π_{A_{0}}η′_{n+1} = ∑_{i=1}^{n} (η′_{n+1}, ξ_{i})_{L^{2}} ξ_{i},

we have

∥Π_{A_{0}}η′_{n+1}∥_{L^{2}} ≤ C\sqrt{λ} and ∥Π_{A_{0}}^{⊥}η′_{n+1}∥_{L^{2}} ≥ 1 − C\sqrt{λ}. (6.13)

Thus, inequalities (6.9), (6.10), and (6.13) give the required estimates for {Π_{A_{0}}η′_{i}}_{i=1}^{n+1} and completes the proof of Lemma 6.12.

6.5. A lower bound for the first non-small eigenvalue. Lastly, we compute a lower bound for the (n + 1)-st eigenvalue of d_{A}^{+}d_{A}^{+,∗}. Recall that η_{n+1} = η_{n+1}(A) is the L^{2}-unit eigenvector associated to the (n + 1)-st eigenvalue μ_{n+1}[A] of d_{A}^{+}d_{A}^{+,∗}.

Lemma 6.14. Continue the above notation. Then there are positive constants C = C(∥F_{A}∥_{L^{2}}, ∥F_{A}^{+}∥_{L^{∞}}, ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}, ν_{2}[A_{0}]) and λ_{0} = λ_{0}(C) such that for all λ ∈ (0, λ_{0}],

μ_{n+1}[A] ≥ ν_{2}[A_{0}] − C\sqrt{λ}.

Proof. Recall that η′_{n+1} = χ_{0}η_{n+1} ∈ Γ(Λ^{+} ⊗ 𝔤_{E_{0}}). Therefore, via the identity (6.7), Lemma 5.8, our hypotheses, and the fact that ∥d_{A}^{+,∗}η_{n+1}∥_{L^{2}} = \sqrt{μ_{n+1}[A]}∥η_{n+1}∥_{L^{2}}, we get

∥d_{A_{0}}^{+,∗}η′_{n+1}∥_{L^{2}} ≤ (∥dχ_{0}∥_{L^{2}} + ∥χ_{0}a∥_{L^{2}}) ∥η_{n+1}∥_{L^{∞}} + ∥d_{A}^{+,∗}η_{n+1}∥_{L^{2}}
≤ c\sqrt{λ}∥η_{n+1}∥_{L^{∞}} + \sqrt{μ_{n+1}[A]}∥η_{n+1}∥_{L^{2}}.

According to Lemma 6.7 we have

∥η_{n+1}∥_{L^{∞}} ≤ c(1 + ∥F_{A}∥_{L^{2}})^{2}(1 + ∥F_{A}^{+}∥_{L^{∞}})^{5/2}(1 + μ_{n+1}[A])(1 + \sqrt{μ_{n+1}[A]})∥η_{n+1}∥_{L^{2}}
≤ C∥η_{n+1}∥_{L^{2}},

using the upper bound for μ_{n+1}[A] in Lemma 6.10. Hence, we obtain

∥d_{A_{0}}^{+,∗}η′_{n+1}∥_{L^{2}} ≤ C\sqrt{λ} + \sqrt{μ_{n+1}[A]}.

In the other direction, we have the eigenvalue estimate ∥d_{A_{0}}^{+,∗}v∥_{L^{2}} ≥ \sqrt{ν_{2}[A_{0}]}∥v∥_{L^{2}} for all v ∈ (Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗})^{⊥} ⊂ L^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}), and so by the orthogonal projection estimates (6.13),

∥d_{A_{0}}^{+,∗}η′_{n+1}∥_{L^{2}} ≥ \sqrt{ν_{2}[A_{0}]}∥Π_{A_{0}}^{⊥}η′_{n+1}∥_{L^{2}} ≥ \sqrt{ν_{2}[A_{0}]}(1 − Cλ).

Combining these upper and lower L^{2} bounds for d_{A_{0}}^{+,∗}η′_{n+1} yields

\sqrt{ν_{2}[A_{0}]} (1 − C\sqrt{λ}) ≤ C\sqrt{λ} + \sqrt{μ_{n+1}[A]},

and this gives the lower bound on μ_{n+1}[A] for small enough λ.

Proof of Theorem 6.1. Combine Lemmas 6.10 and 6.14.

6.6. Variation of eigenvalues with the connection. Our final task is to estimate the variation in the eigenvalues μ_{i}[A] of d_{A}^{+}d_{A}^{+,∗} with the connection A.

Lemma 6.15. Let X be a C^{∞}, closed, oriented, Riemannian four-manifold. Then there is a positive constant c with the following significance. Let E be a rank-two, Hermitian vector bundle over X, let A be an orthogonal L_{4}^{2} connection on 𝔤_{E}, and let 0 ≤ μ_{1}[A] ≤ μ_{2}[A] ≤ ⋯ be the eigenvalues of d_{A}^{+}d_{A}^{+,∗} on L^{2}(Λ^{+} ⊗𝔤_{E}), repeated according to their multiplicity. Then, for any integer i ≥ 1 and a ∈ L_{4}^{2}(Λ^{1} ⊗ 𝔤_{E}),

(1) μ_{i}[A + a] ≤ μ_{i}[A](1 + C∥a∥_{L^{4}(X)} + C∥a∥_{L^{4}(X)}^{2}) + C∥a∥_{L^{4}(X)}(1 + ∥a∥_{L^{4}(X)}),
(2)  μ_{i}[A] ≤ μ_{i}[A + a](1 + C∥a∥_{L^{4}(X)} + C∥a∥_{L^{4}(X)}^{2}) + C∥a∥_{L^{4}(X)}(1 + ∥a∥_{L^{4}(X)}),

where C[A] := c(1 + ∥F_{A}^{+}∥_{L^{∞}}).

Proof. Recall that by Lemma 6.8

μ_{i}[A + a] = inf_{η∈[η_{1}[A+a],...,η_{i−1}[A+a]]^{⊥}} \frac{∥d_{A+a}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}} , i ∈ ℕ,

where the infimum is taken over all η ∈ L_{1}^{2}(Λ^{+} ⊗ 𝔤_{E}) which are L^{2} orthogonal to the span of η_{1}[A + a], …, η_{i−1}[A + a] in L^{2}(Λ^{+} ⊗ 𝔤_{E}), the first i − 1 eigenvectors of d_{A+a}^{+}d_{A+a}^{+,∗} on L^{2}(Λ^{+} ⊗ 𝔤_{E}). Since d_{A+a}^{+,∗}η = d_{A}^{+,∗}η + ∗(a ∧ η) we have, for any η ∈ L_{1}^{2}(Λ^{+} ⊗ 𝔤_{E}),

∥d_{A+a}^{+,∗}η∥_{L^{2}}^{2} = ∥d_{A}^{+,∗}η∥_{L^{2}}^{2} + ∥ ∗ (a ∧ η)∥_{L^{2}}^{2} + 2(d_{A}^{+,∗}η, ∗(a ∧ η))_{L^{2}}
≤ ∥d_{A}^{+,∗}η∥_{L^{2}}^{2}(1 + ∥a∥_{L^{4}}) + (1 + ∥a∥_{L^{4}})∥a∥_{L^{4}}∥η∥_{L^{4}}^{2}.

But Lemmas 6.5 and 6.6 imply that

∥η∥_{L^{4}} ≤ c∥η∥_{L_{1,A}^{2}} ≤ c(1 + ∥F_{A}^{+}∥_{L^{∞}})^{1/2}(∥d_{A}^{+,∗}η∥_{L^{2}} + ∥η∥_{L^{2}}),

for any η ∈ L_{1}^{2}(Λ^{+} ⊗ 𝔤_{E}) and so

∥d_{A+a}^{+,∗}η∥_{L^{2}}^{2} ≤ ∥d_{A}^{+,∗}η∥_{L^{2}}^{2}(1 + ∥a∥_{L^{4}}) + C[A](1 + ∥a∥_{L^{4}})∥a∥_{L^{4}}(∥d_{A}^{+,∗}η∥_{L^{2}} + ∥η∥_{L^{2}})^{2}

≤ ∥d_{A}^{+,∗}η∥_{L^{2}}^{2}(1 + C[A]∥a∥_{L^{4}} + C[A]∥a∥_{L^{4}}^{2}) + C[A](1 + ∥a∥_{L^{4}})∥a∥_{L^{4}}∥η∥_{L^{2}}^{2},

where C[A] = c(1 + ∥F_{A}^{+}∥_{L^{∞}}). By Lemma 6.8 and the preceding estimate we see that

μ_{i}[A + a] = inf_{η∈[η_{1}[A+a],...,η_{i−1}[A+a]]^{⊥}} \frac{∥d_{A+a}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}}

≤ inf_{η∈[η_{1}[A+a],...,η_{i−1}[A+a]]^{⊥}} \frac{∥d_{A}^{+,∗}η∥_{L^{2}}^{2}}{∥η∥_{L^{2}}^{2}} (1 + C[A]∥a∥_{L^{4}} + C[A]∥a∥_{L^{4}}^{2})

+ C[A](1 + ∥a∥_{L^{4}})∥a∥_{L^{4}}

≤ μ_{i}[A](1 + C[A]∥a∥_{L^{4}} + C[A]∥a∥_{L^{4}}^{2}) + C[A](1 + ∥a∥_{L^{4}})∥a∥_{L^{4}},

and this gives the upper bound for μ_{i}[A + a] in terms of μ_{i}[A]. Interchanging the roles of A and A + a gives the upper bound for μ_{i}[A] in terms of μ_{i}[A + a].

7. Existence of gluing maps and obstruction sections

In this section we prove the assertions of Theorem 1.3 concerning the existence and smoothness of the gluing map γ and obstruction section χ, but not the assertions concerning the existence of continuous extensions of these maps to the Uhlenbeck compactification of the space of gluing data, Ḡl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}); the latter Uhlenbeck continuity assertions are taken up in §8.

7.1. The extended anti-self-dual equations. As we saw in §2.6, we need to allow for the presence of cokernel obstructions to deformation in the parameterized moduli space of background connections contained in the gluing data and, thus, small-eigenvalue obstructions to gluing. In this subsection we set up a version of the ‘extended anti-self-dual equations’ which we shall use to take account of such obstructions. Though similar setups are described in [5], [7], [8], [25], [49], [54], and [56], they are not identical so we shall describe our version in some detail.

Let A be an orthogonal connection on an SO(3) bundle 𝔤_{E} = 𝔰𝔲(E) over X, where E is a Hermitian, rank-two vector bundle over X. Our first task is to obtain existence and global estimates for solutions a ∈ L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}) to the quasi-linear equation,

d_{A}^{+}a + (a ∧ a)^{+} = w and a = d_{A}^{+,∗}v, (7.1)

where v ∈ (C^{0} ∩ L_{2}^{♯,2})(Λ^{+} ⊗ 𝔤_{E}) and w ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}). (For the definition and properties of the L^{♯} and related families of Sobolev norms, see [12, §4].) As in [52], [54], [56], we use a = d_{A}^{+,∗}v to rewrite the first-order quasi-linear equation (7.1) for a in terms of w as

d_{A}^{+}d_{A}^{+,∗}v + (d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} = w, (7.2)

to give a second-order, quasi-linear elliptic equation for v in terms of w. For the reader familiar with the construction of gluing maps in [8], the current section gives a generalization of the treatment in [8, §7.2.2 & 7.2.3], where existence of solutions (in certain cases) to the anti-self-dual equation is established.

As in [54], [56], we allow for the possibility that the Laplacian d_{A}^{+}d_{A}^{+,∗} has small eigenvalues. From §6 we know that this occurs when [A] is close, in the Uhlenbeck topology, to a point [A_{0}, x] ∈ B_{κ}^{w}(X) × Sym^{ℓ}(X), where d_{A_{0}}^{+}d_{A_{0}}^{+,∗} has a positive-dimensional kernel. Thus, in place of the anti-self-dual equation,

F^{+}(A + d_{A}^{+,∗}v) = 0, (7.3)

for v ∈ L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}), we follow [54], [56] and consider the pair of equations

Π_{A,μ}^{⊥} · F^{+}(A + d_{A}^{+,∗}v) = 0, (7.4)

Π_{A,μ} · F^{+}(A + d_{A}^{+,∗}v) = 0, (7.5)

for solutions v ∈ L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}) satisfying the constraint Π_{A,μ}v = 0; equation (7.4) is called the extended anti-self-dual equation. Here, Π_{A,μ} is the L^{2}-orthogonal projection from L^{2}(Λ^{+} ⊗ 𝔤_{E}) onto the finite-dimensional subspace spanned by the eigenvectors of d_{A}^{+}d_{A}^{+,∗} with eigenvalues in [0, μ], where μ is a positive constant (an upper bound for the small eigenvalues), and Π_{A,μ}^{⊥} := id − Π_{A,μ}. Because of the small-eigenvalue problem, we shall only explicitly solve equation (7.4); the true solutions to equation (7.3) are then simply the solutions to equation (7.4) which also satisfy equation (7.5).

Equation (7.4) may be rewritten as

Π_{A,μ}^{⊥}(d_{A}^{+}d_{A}^{+,∗}v + (d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+}) = −Π_{A,μ}^{⊥}F_{A}^{+}, (7.6)

where again v ∈ L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}) satisfies the constraint Π_{A,μ}v = 0. For the purposes of the regularity theory of §4, it is useful to rewrite equation (7.6) as

d_{A}^{+}d_{A}^{+,∗}v + (d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} = Π_{A,μ}(d_{A}^{+}d_{A}^{+,∗}v + (d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+}) − Π_{A,μ}^{⊥}F_{A}^{+}. (7.7)

On the other hand, for the purposes of applying Banach-space fixed theory to solve the non-linear equation (7.6), it is convenient to write

v = G_{A,μ}ξ, for some ξ ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{E}),

where G_{A,μ} : L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{E}) → L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}) is the Green’s operator for the Laplacian

d_{A}^{+}d_{A}^{+,∗} : Π_{A,μ}^{⊥} · L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}) → Π_{A,μ}^{⊥} · L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{E}).

Thus, d_{A}^{+}d_{A}^{+,∗}G_{A,μ} = Π_{A,μ}^{⊥} and Π_{A,μ}v = Π_{A,μ}G_{A,μ}ξ = 0 for any ξ ∈ L_{k−1}^{2}(Λ^{+} ⊗ 𝔤_{E}). Therefore, it suffices to solve for ξ such that

ξ + (d_{A}^{+,∗}G_{A,μ}ξ ∧ d_{A}^{+,∗}G_{A,μ}ξ)^{+} = −F_{A}^{+}, (7.8)

because, noting that Π_{A,μ}^{⊥}ξ = Π_{A,μ}^{⊥}d_{A}^{+}d_{A}^{+,∗}G_{A,μ}ξ, the resulting v = G_{A,μ}ξ solves equation (7.6), namely

Π_{A,μ}^{⊥}(ξ + (d_{A}^{+,∗}G_{A,μ}ξ ∧ d_{A}^{+,∗}G_{A,μ}ξ)^{+} + F_{A}^{+}) = 0,

as well as the constraint Π_{A,μ}v = 0.

Noting that Π_{A,μ}d_{A}^{+}d_{A}^{+,∗}G_{A,μ}ξ = 0, the solution ξ to (7.8) solves equation (7.5), namely

Π_{A,μ}((d_{A}^{+,∗}G_{A,μ}ξ ∧ d_{A}^{+,∗}G_{A,μ}ξ)^{+} + F_{A}^{+}) = 0,

if and only if

Π_{A,μ}ξ = 0. (7.9)

For applications of Banach-space fixed-point theory, the integral form (7.8) of the extended anti-self-dual equation is a little more convenient, while for the purposes of regularity theory and localization, we use the “differential” form (7.7).

7.2. Global estimates: linear theory. The results we need are treated in detail in [12], so we just summarize the main conclusions we need here; the L^{♯} and related families of Sobolev norms are defined in [12, §4].

Lemma 7.1. [12, Lemma 5.9] Let X be a closed, oriented four-manifold with metric g. Then there is a constant c with the following significance. Let E be a Hermitian, rank-two vector bundle over X and let A be an orthogonal L_{4}^{2} connection on 𝔤_{E} with curvature F_{A}. Then the following estimate holds for any v ∈ L_{2}^{♯,2}(Λ^{+} ⊗ 𝔤_{E}):

∥v∥_{L_{2,A}^{2}(X)} + ∥v∥_{C^{0}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})^{2}(1 + ∥F_{A}^{+}∥_{C^{0}(X)})^{2}
× (∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{♯,2}(X)} + ∥v∥_{L^{2}(X)}).

Corollary 7.2. [12, Corollary 5.10] Continue the hypotheses of Lemma 7.1. Then:

∥d_{A}^{+,∗}v∥_{L_{1,A}^{2}(X)} ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})^{2}(1 + ∥F_{A}^{+}∥_{C^{0}(X)})^{2}
× (∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{♯,2}(X)} + ∥v∥_{L^{2}(X)}).

Note that if a ∈ L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}) is L^{2}-orthogonal to Ker d_{A}^{+}, so that a = d_{A}^{+,∗}v for some v ∈ L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}), then the estimate of Corollary 7.2 can be written in the more familiar form (see [12, Eq. (5.10)])

\begin{aligned}∥a∥_{L_{1,A}^{2}(X)} & ≤ c(1 + ∥F_{A}∥_{L^{2}(X)})^{2}(1 + ∥F_{A}^{+}∥_{C^{0}(X)})^{2} \\ & × (∥d_{A}^{+}a∥_{L^{♯,2}(X)} + μ^{−1/2}∥a∥_{L^{2}(X)}), \end{aligned} (7.10)

where μ = ν_{2}[A] > 0 and we make use of the eigenvalue estimate ∥v∥_{L^{2}} ≤ μ^{−1/2}∥d_{A}^{+,∗}v∥_{L^{2}}; the term d_{A}^{+}a above can be replaced by (d_{A}^{+,∗}d_{A}^{+})a without changing the estimate constants. More generally, if μ is any positive constant and Π_{A,μ}v = 0, then the estimate (7.10) continues to hold.

Fix a positive constant μ. Given an L_{4}^{2} orthogonal connection A on 𝔤_{E}, we define a partial right inverse

P_{A,μ} := d_{A}^{+,∗} (d_{A}^{+}d_{A}^{+,∗})^{−1} · Π_{A,μ}^{⊥} : L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}) → L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}),

to the operator

d_{A}^{+} : L_{1}^{2}(Λ^{1} ⊗ 𝔤_{E}) → L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}).

We have the following estimates for P_{A,μ}:

Proposition 7.3. Let X be a closed, oriented, smooth four-manifold with Riemannian metric g. Let A be an L_{4}^{2} connection on 𝔤_{E} and let μ be a positive constant. Then there is a positive constant C(∥F_{A}^{+}∥_{L^{∞}}, ∥F_{A}^{−}∥_{L^{2}}, μ) such that for all ξ ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}),

(1)  ∥P_{A,μ}ξ∥_{L^{2}} ≤ C∥ξ∥_{L^{4/3}},
(2)  ∥P_{A,μ}ξ∥_{L_{1,A}^{2}} ≤ C∥ξ∥_{L^{♯,2}}.

Proof. We set v = G_{A,μ}ξ, so that Π_{A,μ}v = 0 with

d_{A}^{+}d_{A}^{+,∗}v = Π_{A,μ}^{⊥}ξ and a = d_{A}^{+,∗}v = P_{A,μ}ξ.

We first consider the L^{2} estimate for a. We integrate by parts to get

∥a∥_{L^{2}}^{2} = ∥d_{A}^{+,∗}v∥_{L^{2}}^{2} = (d_{A}^{+}d_{A}^{+,∗}v, v)_{L^{2}} = (Π_{A,μ}^{⊥}ξ, v)_{L^{2}} = (ξ, v)_{L^{2}}
≤ ∥ξ∥_{L^{4/3}}∥v∥_{L^{4}} ≤ c∥ξ∥_{L^{4/3}}∥v∥_{L_{1,A}^{2}}.

From Lemma 6.6 and the eigenvalue estimate ∥v∥_{L^{2}} ≤ μ^{−1/2}∥d_{A}^{+,∗}v∥_{L^{2}}, we have

∥v∥_{L_{1,A}^{2}} ≤ C(∥d_{A}^{+,∗}v∥_{L^{2}} + ∥v∥_{L^{2}}) ≤ C(∥d_{A}^{+,∗}v∥_{L^{2}} + μ^{−1/2}∥d_{A}^{+,∗}v∥_{L^{2}}) ≤ C∥d_{A}^{+,∗}v∥_{L^{2}}.

Therefore,

∥d_{A}^{+,∗}v∥_{L^{2}}^{2} ≤ C∥ξ∥_{L^{4/3}}∥d_{A}^{+,∗}v∥_{L^{2}},

and so

∥d_{A}^{+,∗}v∥_{L^{2}} ≤ C∥ξ∥_{L^{4/3}},

which gives the desired L^{2} estimate for a = d_{A}^{+,∗}v. We can also combine the preceding estimates, noting that we can choose to either retain or drop the projection Π_{A,μ}^{⊥} as above, to give

\begin{aligned}∥v∥_{L_{1,A}^{2}} & ≤ C∥Π_{A,μ}^{⊥}ξ∥_{L^{4/3}}, \\ ∥v∥_{L_{1,A}^{2}} & ≤ C∥ξ∥_{L^{4/3}}, \end{aligned} (7.11)

estimates we shall later need.

We next derive the L_{1,A}^{2} estimate for a. Since a = d_{A}^{+,∗}v, then

∥a∥_{L_{1,A}^{2}} ≤ ∥d_{A}^{+,∗}v∥_{L_{1,A}^{2}}.

Using integration by parts, we see that

∥d_{A}^{+,∗}v∥_{L^{2}}^{2} = (d_{A}^{+,∗}v, d_{A}^{+,∗}v)_{L^{2}} = (d_{A}^{+}d_{A}^{+,∗}v, v)_{L^{2}}
≤ ∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{2}}∥v∥_{L^{2}} ≤ \frac{1}{2}(∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{2}}^{2} + ∥v∥_{L^{2}}^{2}),

and so

∥d_{A}^{+,∗}v∥_{L^{2}} ≤ \frac{1}{\sqrt{2}}(∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{2}} + ∥v∥_{L^{2}}). (7.12)

By combining Corollary 7.2 with equation (7.12) and noting that d_{A}^{+}d_{A}^{+,∗}v = Π_{A,μ}^{⊥}ξ = ξ − Π_{A,μ}ξ, we obtain

∥a∥_{L_{1,A}^{2}} ≤ C(∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{♯,2}} + ∥v∥_{L^{2}})
= C(∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{♯,2}} + μ^{−1}∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{2}})
≤ C∥Π_{A,μ}^{⊥}ξ∥_{L^{♯,2}}
≤ C∥ξ∥_{L^{♯,2}}, (by Lemma 6.7 and [12, Lemma 4.1])

which gives the required L_{1,A}^{2} estimate for a. Of course, the preceding argument also yields

\begin{aligned}∥v∥_{C^{0}∩L_{2,A}^{2}} & ≤ C∥Π_{A,μ}^{⊥}ξ∥_{L^{♯,2}}, \\ ∥v∥_{C^{0}∩L_{2,A}^{2}} & ≤ C∥ξ∥_{L^{♯,2}}, \end{aligned} (7.13)

estimates we shall later need.

7.3. Global estimates: non-linear theory. We now extend our global estimates in §7.2 for the linear operator d_{A}^{+}d_{A}^{+,∗} to estimates for the non-linear operator d_{A}^{+}d_{A}^{+,∗} + (d_{A}^{+,∗}(·) ∧ d_{A}^{+,∗}(·))^{+}. By adapting the arguments of [52], [55], [56] we first obtain global C^{0} ∩ L_{2,A}^{2} estimates for solutions v to equation (7.2) and thus global L_{1,A}^{2} estimates for solutions a = d_{A}^{+,∗}v to equation (7.1). These global estimates have the property that the constant C is uniform with respect to the point [A] ∈ B_{E}(X) in the sense that C depends on A only through ∥F_{A}^{+}∥_{L^{∞}}, ∥F_{A}∥_{L^{2}}, and μ, a choice of positive, small-eigenvalue cutoff constant for the Laplacian d_{A}^{+}d_{A}^{+,∗} on L^{2}(Λ^{+} ⊗ 𝔤_{E}). We will make frequent use of the following a priori estimate for L_{2}^{2} solutions to equation (7.2).

Lemma 7.4. Let X be a closed, oriented, smooth four-manifold with Riemannian metric g, and let M, Λ be positive constants. Then there are positive constants ε, C such that the following holds. Let A be an L_{4}^{2} connection on 𝔤_{E} with ∥F_{A}^{+}∥_{L^{∞}} ≤ M, ∥F_{A}∥_{L^{2}} ≤ M, and let μ be a positive constant such that Λ^{−1} ≤ μ ≤ Λ. Let v ∈ (C^{0} ∩ L_{2}^{2})(Λ^{+} ⊗ 𝔤_{E}) and w ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}) solve (7.2), with Π_{A,μ}v = 0, and suppose that ∥v∥_{L_{2,A}^{2}(X)} < ε. Then

(1)  ∥v∥_{L_{1,A}^{2}(X)} ≤ C∥w∥_{L^{4/3}(X)},
(2)  ∥v∥_{L_{2,A}^{2}(X)} + ∥v∥_{L^{∞}(X)} ≤ C∥w∥_{L^{♯,2}(X)}

Proof. Note that estimate (7.11) yields

∥v∥_{L_{1,A}^{2}} ≤ C∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{4/3}},

while from equation (7.2) and the Sobolev multiplications L_{1,A}^{2} × L^{2} → L^{4/3} (with constant depending only on the metric) we have

∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{4/3}} ≤ ∥(d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+}∥_{L^{4/3}} + ∥w∥_{L^{4/3}}
≤ c∥d_{A}^{+,∗}v∥_{L_{1,A}^{2}}∥d_{A}^{+,∗}v∥_{L^{2}} + ∥w∥_{L^{4/3}},

and so,

∥v∥_{L_{1,A}^{2}} ≤ C (∥v∥_{L_{2,A}^{2}}∥v∥_{L_{1,A}^{2}} + ∥w∥_{L^{4/3}}) .

Thus, taking ε ≤ 1/(2C), the first estimate follows by rearrangement.

Hölder’s inequality and our elliptic estimate in Lemma 7.1 for the Laplacian d_{A}^{+}d_{A}^{+,∗} yields

∥v∥_{L_{2,A}^{2}} + ∥v∥_{L^{∞}} ≤ C∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{♯,2}} + ∥v∥_{L^{2}},

while from equation (7.2) and the Sobolev multiplications L_{1,A}^{2} × L_{1,A}^{2} → L^{♯,2} (again with constant depending only on the metric) we have

∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{♯,2}} ≤ ∥(d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+}∥_{L^{♯,2}} + ∥w∥_{L^{♯,2}}
≤ c∥d_{A}^{+,∗}v∥_{L_{1,A}^{2}}^{2} + ∥w∥_{L^{♯,2}},

and so,

∥v∥_{L_{2,A}^{2}} + ∥v∥_{L^{∞}} ≤ C (∥v∥_{L_{2,A}^{2}}^{2} + ∥w∥_{L^{♯,2}}) .

Thus, taking ε ≤ 1/2C, the second estimate follows by rearrangement.

7.4. Existence of solutions to the extended anti-self-dual equation. We now turn to the question of existence and uniqueness of solutions to equation (7.4) or one of its equivalent forms, namely equations (7.6) or (7.7). We first recall the following elementary fixed-point result [8, Lemma 7.2.23].

Lemma 7.5. Let q : 𝔅 → 𝔅 be a continuous map on a Banach space 𝔅 with q(0) = 0 and

∥q(x_{1}) − q(x_{2})∥ ≤ K (∥x_{1}∥ + ∥x_{2}∥) ∥x_{1} − x_{2}∥

for some positive constant K and all x_{1}, x_{2} in 𝔅. Then for each y in 𝔅 with ∥y∥ < 1/(10K) there is a unique x in 𝔅 such that ∥x∥ ≤ 1/(5K) and x + q(x) = y.

With the aid of Lemma 7.5, we obtain existence and uniqueness for solutions to equation (7.4).

Proposition 7.6. Let (X, g) be a closed, oriented, C^{∞} Riemannian four-manifold, and let M, Λ be positive constants. Then there are positive constants ε, C such that the following holds. Let E be a Hermitian, rank-two vector bundle and let A be an L_{4}^{2} orthogonal connection on 𝔤_{E} with ∥F_{A}^{+}∥_{L^{∞}} ≤ M, ∥F_{A}∥_{L^{2}} ≤ M, and let μ be a positive constant such that Λ^{−1} ≤ μ ≤ Λ. Suppose that ∥F_{A}^{+}∥_{L^{♯,2}(X)} < ε. Then there is a unique solution v ∈ (C^{0} ∩ L_{2}^{2})(Λ^{+} ⊗ 𝔤_{E}) to equation (7.4) such that Π_{A,μ}v = 0 and

(1)  ∥v∥_{L_{1,A}^{2}(X)} ≤ C∥F_{A}^{+}∥_{L^{4/3}(X)},
(2)  ∥v∥_{L_{2,A}^{2}(X)} + ∥v∥_{L^{∞}(X)} ≤ C∥F_{A}^{+}∥_{L^{♯,2}(X)}.

Moreover, if A is an L_{l}^{2} connection, for any l ≥ 4, then the solution v is contained in L_{l+1}^{2}(Λ^{+} ⊗ 𝔤_{E}).

Proof. We try to solve equation (7.6) for solutions of the form v = G_{A,μ}ξ, where ξ ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}) and G_{A,μ} is the Green’s operator for the Laplacian d_{A}^{+}d_{A}^{+,∗} on Ran Π_{A,μ}^{⊥}. Therefore, as explained in §7.1, we seek solutions ξ to equation (7.8). We now apply Lemma 7.5 to equation (7.8), with 𝔅 = L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}), choosing y = −F_{A}^{+} and q(ξ) = (d_{A}^{+,∗}G_{A,μ}ξ ∧ d_{A}^{+,∗}G_{A,μ}ξ)^{+}, so our goal is to solve

ξ + q(ξ) = −F_{A}^{+} on L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}).

For any ξ_{1}, ξ_{2} ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}), the estimate in Proposition 7.3 yields

∥d_{A}^{+,∗}G_{A,μ}ξ_{i}∥_{L_{1,A}^{2}} ≤ C∥ξ_{i}∥_{L^{♯,2}}.

Therefore, applying the preceding estimates, Hölder’s inequality, Lemma 6.5, and the L^{♯}- family of embedding and multiplication results of [12, Lemmas 4.1 & 4.3], we obtain

∥q(ξ_{1}) − q(ξ_{2})∥_{L^{♯,2}} ≤ K_{2} (∥ξ_{1}∥_{L^{♯,2}} + ∥ξ_{2}∥_{L^{♯,2}}) ∥ξ_{1} − ξ_{2}∥_{L^{♯,2}},

where K_{2} = cZ_{2} with c a universal constant and Z_{2} the constant of Proposition 7.3.

Thus, provided ε ≤ (10K_{2})^{−1}, we have ∥F_{A}^{+}∥_{L^{♯,2}} < (10K_{2})^{−1}, and Lemma 7.5 implies that there is a unique solution ξ ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}) to equation (7.8) such that ∥ξ∥_{L^{♯,2}} ≤ (5K_{2})^{−1}. The L^{4/3} and L^{♯,2} estimates for ξ,

∥ξ∥_{L^{4/3}} ≤ 2∥F_{A}^{+}∥_{L^{4/3}} and ∥ξ∥_{L^{♯,2}} ≤ 2∥F_{A}^{+}∥_{L^{♯,2}}, (7.14)

follow from Proposition 7.3 and a standard rearrangement argument. To obtain the stated L_{1,A}^{2} and L_{2,A}^{2} estimates for v, observe that

d_{A}^{+}d_{A}^{+,∗}v = d_{A}^{+}d_{A}^{+,∗}G_{A,μ}ξ = Π_{A,μ}^{⊥}ξ, (7.15)

so that the L_{1,A}^{2} bound is obtained from

∥v∥_{L_{1,A}^{2}} ≤ C∥ξ∥_{L^{4/3}} (by estimate (7.11))
≤ C∥F_{A}^{+}∥_{L^{4/3}}, (by estimate (7.14)).

The C^{0} ∩ L_{2,A}^{2} estimate similarly follows, as

∥v∥_{C^{0}∩L_{2,A}^{2}} ≤ C∥ξ∥_{L^{♯,2}} (by estimate (7.13))
≤ C∥F_{A}^{+}∥_{L^{♯,2}} (by estimate (7.14)).

The final regularity assertion follows from Proposition 4.22, our regularity result for gluing solutions to the (extended) anti-self-dual equation.

7.5. Proof of main theorem, without Uhlenbeck continuity assertions. At this stage, we are ready to complete the proof of at least part of Theorem 1.3, namely the existence of gluing and obstruction maps, but not the Uhlenbeck continuity assertions for these maps, which requires considerably more work and is taken up in §8. On the other hand, the proof of this first part of Theorem 1.3 is an essentially formal, almost immediate consequence of Proposition 7.6.

Proof of Theorem 1.3, without Uhlenbeck continuity assertions. According to Proposition 7.6 there are well-defined, G_{E}-equivariant maps δ and φ from the open subset of A_{κ}^{w}(X) × C,

\begin{aligned}A_{κ}^{w}(X, C; M, μ, ε) & := \{(A, g) ∈ A_{κ}^{w}(X) × C : \\ & Spec(d_{A}^{+,g}d_{A}^{+,g,∗}) ⊂ [0, \frac{1}{2}μ) ∪ (μ, ∞), \\ & ∥F_{A}^{+,g}∥_{L^{♯,2}} < ε, and ∥F_{A}^{+,g}∥_{L^{∞}} < M\}, \\ B_{κ}^{w}(X, C; M, μ, ε) & := A_{κ}^{w}(X, C; M, μ, ε)/G_{E}, \end{aligned} (7.16)

given by

δ : (A, g) ↦ (A + d_{A}^{+,g,∗}v, g),
φ : (A, g) ↦ Π_{A,μ}F^{+,g}(A + d_{A}^{+,g,∗}v),

into A_{κ}^{w}(X) × C and L^{2}(X, Λ^{+,g} ⊗ 𝔤_{E}), respectively. These descend to a map and a section,

\begin{matrix}δ : B_{κ}^{w}(X, C; M, μ, ε) → B_{κ}^{w}(X) × C, \\ φ : B_{κ}^{w}(X, C; M, μ, ε) → 𝔙_{μ}, \end{matrix} (7.17)

where 𝔙_{μ} → B_{κ}^{w}(X, C; μ) is the continuous, finite-rank, pseudovector bundle

𝔙_{μ} := {(A, g, w) : (A, g) ∈ A_{κ}^{w}(X, C; μ) and w ∈ Ran Π_{A,g,μ}}/G_{E},

where we recall that Ran Π_{A,g,μ} ⊂ L_{k−1}^{2}(X, Λ^{+,g} ⊗ 𝔤_{E}). Here, the spaces A_{κ}^{w}(X, C; μ) and B_{κ}^{w}(X, C; μ) are defined by simply omitting the conditions in definition (7.16) involving M and ε. The map δ and section φ have the desired property that

δ(φ^{−1}(0)) ⊂ M_{κ}^{w}(X, C),

and are smooth respect to the quotient L_{k}^{2} topology upon restriction to B_{κ}^{w,∗}(X): smoothness follows by the same argument as used in the proof of smoothness in the case of the Banachspace inverse function theorem [1], where the existence of an inverse is established via a contraction-mapping argument, just as we employ here.

The gluing map, gluing-data obstruction bundle, and gluing-data obstruction section are then defined by

γ = δ ∘ γ′ : Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) → B_{κ}^{w}(X) × C,
χ = φ ∘ γ : Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) → Ξ,
Ξ = γ^{∗}𝔙_{μ},

where the open neighborhood U_{0} ⊂ M_{κ_{0}}^{w}(X, C) is chosen according to Remark 2.11. The estimates in §5 and §6 ensure that the image of γ′, when γ′ is restricted to Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}), is contained in B_{κ}^{w,∗}(X, C; M, μ, ε): in particular, that the a priori estimates on F_{A}^{+,g} are obeyed (by Corollary 5.13) and that the spectrum of d_{A_{0}}^{+,g}d_{A_{0}}^{+,∗,g} obeys the required open constraints for a suitable μ (by Theorem 6.1). We can choose μ to be the infimum of \frac{1}{2}ν_{2}[A_{0}, g] (the least positive eigenvalue of the Laplacian d_{A_{0}}^{+,g}d_{A_{0}}^{+,∗,g}), as [A_{0}, g] ∈ B_{κ_{0}}^{w}(X) × C ranges over the space of gauge-equivalence classes of background connections (and metrics in C) contained in Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}).

Remark 7.7. The rank of 𝔙_{μ} varies over B_{κ}^{w}(X, C; M, μ, ε), so that is why we only refer to it as being a ‘pseudovector bundle’. However, the construction of Gl_{κ}^{w,+}(X, U_{0}, Σ, λ_{0}) (via the choice of family U_{0} of background connections A_{0}) in §3 and Theorem 6.1 ensure that its image under γ′ is contained in an open subset of B_{κ}^{w}(X, C; M, μ, ε) over which 𝔙_{μ} has constant rank and is thus a vector bundle when restricted to this open subset.

8. Continuity of gluing maps and obstruction sections

Our goal in this section is to prove the assertions in Theorem 1.3 concerning the Uhlenbeck continuity of the gluing map γ and continuity of the Kuranishi obstruction map χ.

As we briefly discussed in §1, the statements of Theorems 3.4.10, 3.4.17, and 3.4.18 in [25] describe a gluing result similar to but less general than our main gluing theorem for antiself-dual connections (see §1 here and [14, §1]). While proofs of [25, Theorems 3.4.10, 3.4.17, & 3.4.18] are attributed by Friedman and Morgan [25] to Taubes (specifically, [52] and [54]), it is nonetheless fair to point out that several important assertions of [25, Theorems 3.4.10, 3.4.17, & 3.4.18] are not proved in [52], [54] (or their sequels [55], [56]). In the preceding four papers of Taubes, the gluing map γ is shown — either directly or by routine arguments or modifications of existing arguments — to have the following properties:

• The gluing construction gives a map γ from Gl_{κ}^{w}(X, U_{0}, K, λ_{0}) into M̄_{κ}^{w}(X, C), which is smooth upon restriction to smooth strata of the gluing-data bundle Gl_{κ}^{w}(X, U_{0}, K, λ_{0}). (Here, K ⋐ Σ is a precompact subset of a smooth stratum Σ of the symmetric product Sym^{ℓ}(X) and C ⊂ Met(X) is a compact subset.)

• Points in M_{κ}^{w}(X, C) are shown to lie in the image of certain iterated gluing maps, whose definition is significantly different from that of γ.

In the versions stated above we have assumed, for simplicity, that the “background connections” A_{0} in Gl_{κ}^{w}(X, U_{0}, K, λ_{0}) have non-vanishing H_{A_{0}}^{2} as the issue of obstructions to gluing — though a complication — is not germane to the main concern of this section, namely Uhlenbeck continuity.

Contrary to the assertions of [25], it is not shown that γ is actually a homeomorphism from Ḡl_{κ}^{w}(X, Ū_{0}, K, λ_{0}) onto an open subset of M̄_{κ}^{w}(X, C), nor is it shown to be a diffeomorphism upon restriction to smooth strata. Though desirable, these properties of γ are not obvious and do not follow by routine modifications of arguments contained in his papers or those, for example, of Donaldson [5], [8] or Mrowka [49]. The other difficulty, which we discussed in §1.1.1, concerns the implicit assumption in [25] that the Taubes perturbation method can always be used to perturb approximately anti-self-dual to anti-self-dual solutions when the standard round metric on S^{4} near the north pole does not match the metric g on X near the gluing sites. While this not a problem when the subsets of the S^{4} moduli spaces being glued in are compact (as in [8], [52], [54]), it becomes a serious issue when the subsets are non-compact (as in [56]) if one wishes — as we do and as is required by [25]

— to both preserve the locally-trivial fiber-bundle structure of the gluing-data spaces and have a topological embedding of the gluing-data spaces (as opposed to a weaker relation, such as a homotopy equivalence as in [56]). This particular difficulty is discussed further in the introduction to §9.

Our goal in this section is to complete the proof of one assertion of [25], namely that the gluing maps extend continuously to the Uhlenbeck closure of the gluing data, Ḡl_{κ}^{w,+}(X, Ū_{0}, Σ̄, λ_{0}), as discussed in §1.1.2. Note that this continuity assertion is considerably stronger than that of [25, Theorem 3.4.10], where Σ has been replaced by a precompact open subset K ⊂ Σ and so the gluing sites are not allowed to approach each other. Our proof will also show that the gluing maps are continuous with respect to bubble-tree limits. A special case of the continuity assertion is treated by Donaldson and Kronheimer in [8, §7.2.8].

To show that our gluing maps are open, it suffices to consider their restrictions to smooth strata: openness then follows from dimension counting and the fact that the gluing maps are local diffeomorphisms onto their images after restriction to smooth strata. These and other properties are proved in [14].

8.1. Statement of continuity results. Let A, A_{0} be connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}}, respectively, over X with c_{1}(E) = c_{1}(E_{0}) and c_{2}(E)−c_{2}(E_{0}) =: ℓ > 0. Though we make no specific use of this fact for now, the case we have in mind is where the point [A] ∈ B_{E}(X, g) lies in an Uhlenbeck neighborhood of the point [A_{0}, x] ∈ B_{E_{0}}(X, g)×Sym^{ℓ}(X) (see Definition 3.1). Let a ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E}) and a_{0} ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}) be solutions to the quasi-linear equations (7.1) on the bundles 𝔤_{E} and 𝔤_{E_{0}}, respectively,

(8.1)  d_{A}^{+}a + (a ∧ a)^{+} = w and a = d_{A}^{+,∗}v,
(8.2)  d_{A_{0}}^{+}a_{0} + (a_{0} ∧ a_{0})^{+} = w_{0} and a_{0} = d_{A_{0}}^{+,∗}v_{0},

where v, w ∈ Γ(X, Λ^{+} ⊗ 𝔤_{E}) and v_{0}, w_{0} ∈ Γ(X, Λ^{+} ⊗ 𝔤_{E_{0}}). The use of w, w_{0} on the righthand-sides of equations (8.1) and (8.2) allows us to give a common solution to a wider range of Uhlenbeck continuity problems than would be possible if we simply chose w = −F_{A}^{+} or w_{0} = F_{A_{0}}^{+}.

We wish to compare the solutions a and a_{0} to equations (8.1) and (8.2) when the connection A is close to (A_{0}, x) in the Uhlenbeck topology and w is close to w_{0} in L_{k−1,loc}^{2} on X \ x. The purpose of this section is to show that a then approaches a_{0} in L_{k,loc}^{2} on X \ x; more precise bounds are given below.

As earlier we use a(A) := d_{A}^{+,∗}v and a_{0} = a(A_{0}) := d_{A_{0}}^{+,∗}v_{0} to rewrite the first-order quasi-linear equations (8.1) and (8.2) for a and a_{0} as

(8.3)  d_{A}^{+}d_{A}^{+,∗}v + (d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} = w,
(8.4)  d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v_{0} + (d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v_{0})^{+} = w_{0},

to give a pair of second-order, quasi-linear elliptic equations for v and v_{0}. In our applications, the terms w and w_{0} will always depend on A and A_{0} in such a way as to preserve gauge invariance. For example, when the Laplacian d_{A}^{+}d_{A}^{+,∗} has no small eigenvalues, we set w(A) := −F_{A}^{+} in equation (8.1) and so we recover the usual anti-self-dual equation. If the Laplacian d_{A}^{+}d_{A}^{+,∗} does have small eigenvalues, we require that Π_{A,μ}v = 0 and set

w(A) := Π_{A,μ}((d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} + F_{A}^{+}) − F_{A}^{+}, (8.5)

and so recover the extended anti-self-dual equation (7.4) or, in detail, (7.6), where Π_{A,μ} is the L^{2}-orthogonal projection from L^{2}(Λ^{+} ⊗ 𝔤_{E}) onto the finite-dimensional subspace spanned by the eigenvectors of d_{A}^{+}d_{A}^{+,∗} with eigenvalues in [0, μ], where μ is a positive constant (an upper bound for the small eigenvalues), usually \frac{1}{2}ν_{2}[A_{0}]: see the small-eigenvalue estimates supplied by Theorem 6.1. Similarly for the connection A_{0}.

We can now state our main results.

Theorem 8.1. Let E, E_{0} be Hermitian vector bundles over a closed, oriented, smooth fourmanifold X with Riemannian metric g, such that c_{1}(E_{0}) = c_{2}(E) and c_{2}(E_{0}) < c_{2}(E). Let A_{0} be an orthogonal L_{k}^{2} connection on 𝔤_{E_{0}} and suppose {A_{α}} is a sequence of L_{k}^{2} connections on 𝔤_{E} which converges to (A_{0}, x) ∈ A_{E_{0}} × Sym^{ℓ}(X) in the sense that

• The sequence of connections A_{α} converges to A_{0} in L_{k,A_{0},loc}^{2} on X \ x, and

• The sequence of measures |F_{A_{α}}|^{2} converges to |F_{A_{0}}|^{2} + 8π^{2} ∑_{x∈x} δ_{x} on X.

We assume that E|_{X\x} = E_{0}|_{X\x}. Suppose v_{0} ∈ L_{k+1}^{2}(Λ^{+}⊗𝔤_{E_{0}}) is the solution to the secondorder quasi-linear equation (8.3) on 𝔤_{E_{0}}, with right-hand side w_{0}, while v_{α} ∈ L_{k+1}^{2}(Λ^{+} ⊗𝔤_{E}) are the solutions to the second-order quasi-linear equation (8.3) on 𝔤_{E}, with right-hand side w_{α}. Suppose that the sequence {w_{α}} converges to w_{0} in L_{k−1,A_{0},loc}^{2}(X \ x). Then the following hold:

• The sequence of solutions v_{α} to equation (8.3) converges to the solution v_{0} to equation (8.4) in L_{k+1,A_{0},loc}^{2} on X \ x,

• The sequence of solutions a_{α} = d_{A_{α}}^{+,∗}v_{α} to equation (8.1) converges to the solution a_{0} = d_{A_{0}}^{+,∗}v_{0} to (8.2) in L_{k,A_{0},loc}^{2} on X \ x,

• The sequence of connections A_{α} + a_{α} converges to A_{0} + a_{0} in L_{k,A_{0},loc}^{2} on X \ x, and • The sequence of measures |F (A_{α} + a_{α})|^{2} converges to |F (A_{0} + a_{0})|^{2} + 8π^{2} ∑_{x∈x} δ_{x} on

X.

The preceding result has the following consequence when applied to the extended antiself-dual equation:

Theorem 8.2. Let E, E_{0} be Hermitian vector bundles over a closed, oriented, smooth fourmanifold X with Riemannian metric g, such that c_{1}(E_{0}) = c_{2}(E) and c_{2}(E_{0}) < c_{2}(E). Let M < ∞, ε > 0 be constants, let A_{0} be an orthogonal L_{k}^{2} connection on 𝔤_{E_{0}} and suppose {A_{α}} is a sequence of L_{k}^{2} connections on 𝔤_{E} such that

(8.6)  sup_{α} ∥F_{A_{α}}^{+}∥_{C^{0}(X)} ≤ M and ∥F_{A_{0}}^{+}∥_{L^{2}(X)}, sup_{α} ∥F_{A_{α}}^{+}∥_{L^{2}(X)} ≤ ε,

and which converges to (A_{0}, x) ∈ A_{E_{0}} × Sym^{ℓ}(X) in the sense that • The sequence of connections A_{α} converges to A_{0} in L_{k,A_{0},loc}^{2} on X \ x, and

• The sequence of measures |F_{A_{α}}|^{2} converges to |F_{A_{0}}|^{2} + 8π^{2} ∑_{x∈x} δ_{x} on X.

We assume that E|_{X\x} = E_{0}|_{X\x} and that ε ≪ 1. Suppose v_{0} ∈ Π_{A_{0},μ}^{⊥} · L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E_{0}}) is the solution to the extended second-order anti-self-dual equation (8.4) on 𝔤_{E_{0}}, while v_{α} ∈ Π_{A_{α},μ}^{⊥} · L_{k+1}^{2}(Λ^{+} ⊗ 𝔤_{E}) are the solutions to the extended second-order anti-self-dual equation (8.3) on 𝔤_{E}, so that

w_{α} := w(A_{α}) = Π_{A_{α},μ}((d_{A_{α}}^{+,∗}v_{α} ∧ d_{A_{α}}^{+,∗}v_{α})^{+} + F_{A_{α}}^{+}) − F_{A_{α}}^{+},
w_{0} := w(A_{0}) = Π_{A_{0},μ}((d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v_{0})^{+} + F_{A_{0}}^{+}) − F_{A_{0}}^{+}.

Then the following hold:

• The sequence of solutions v_{α} to equation (8.3) converges to the solution v_{0} to equation (8.4) in L_{k+1,A_{0},loc}^{2} on X \ x,

• The sequence of terms w_{α} converges to the term w_{0} in L_{k−1,A_{0},loc}^{2} on X \ x,

• The sequence of solutions a_{α} = d_{A_{α}}^{+,∗}v_{α} to equation (8.1) converges to the solution a_{0} = d_{A_{0}}^{+,∗}v_{0} to (8.2) in L_{k,A_{0},loc}^{2} on X \ x,

• The sequence of connections A_{α} + a_{α} converges to A_{0} + a_{0} in L_{k,A_{0},loc}^{2} on X \ x, and • The sequence of measures |F (A_{α} + a_{α})|^{2} converges to |F (A_{0} + a_{0})|^{2} + 8π^{2} ∑_{x∈x} δ_{x} on

X.

Theorem 8.2 has the following important consequence.

Proof of Uhlenbeck continuity assertions of Theorem 1.3. Let us first suppose that the metric g on X is flat near the gluing sites, so that the pre-splicing and splicing maps, γ′ and γ′′, coincide. The assertions in Theorem 1.3 concerning the Uhlenbeck continuity of the gluing map γ and continuity of the Kuranishi obstruction map χ follow immediately from the definition of the splicing map γ′ and Theorem 8.2. For the general case, when the metric g is not necessarily flat near the gluing sites, the argument is the same except that we must now appeal to Theorem 1.1 in order to obtain the required preliminary homeomorphisms between M̄_{κ_{i}}^{⋄,s}(S^{4}, g_{0}) and M̄_{κ_{i}}^{⋄,s}(S^{4}, g̃_{i}).

The remainder of this section is taken up with the proofs of Theorems 8.1 and 8.2.

8.2. Degeneration of solutions to the second-order quasi-linear equation: linear L_{2}^{2} theory. Prior to analyzing the asymptotic behavior of solutions to the full anti-self-dual equation, it is useful to first consider the asymptotic behavior of solutions v ∈ L_{k+1}^{2}(X, Λ^{+}⊗ 𝔤_{E}) to the linear part of the second order, elliptic anti-self-dual equation (7.2), so

d_{A}^{+}d_{A}^{+,∗}v = w, (8.7) where w ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and the point [A] ∈ B_{E}(X, g) converges to the point [A_{0}, x] ∈ B_{E_{0}}(X, g) × Sym^{ℓ}(X).

We need to compare the solutions v to equation (8.7) as A converges to A_{0} and w converges to w_{0} in L_{k−1,A_{0},loc}^{2} on X \ x. For this purpose we use an elementary Leibnitz formula for d_{A}^{+}d_{A}^{+,∗}. We recall from (6.2) that d_{A}^{+}d_{A}^{+,∗} = \frac{1}{2}∇_{A}^{∗}∇_{A} + zeroth-order terms on L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and so we have, schematically, for any function ψ_{0} ∈ C^{∞}(X),

d_{A}^{+}d_{A}^{+,∗}(ψ_{0}v) = ∇^{2}ψ_{0} ⊗ v + ∇ψ_{0} ⊗ ∇_{A}v + ψ_{0}d_{A}^{+}d_{A}^{+,∗}v. (8.8)

The ‘excision’ argument which allows us to compare solutions to the anti-self-dual equation on different bundles E and E_{0} relies on a suitable choice of cutoff functions.

Over X \ x we may write A = A_{0} + (A − A_{0}) on E|_{X\x} = E_{0}|_{X\x}. Though we shall ultimately require a pair of sequences of x-good cutoff functions satisfying the properties of Lemma 6.9, it suffices for now to choose smooth cutoff functions χ_{0} and ψ_{0} on X such that

• supp χ_{0} ⋐ X \ x, supp ψ_{0} ⋐ X \ x,

• χ_{0} = 1 on supp ψ_{0}, and define open subsets of X by

Ω := {x ∈ X : (dψ_{0})_{x} ≠ 0} and U := {x ∈ X : ψ_{0}(x) > 0}.

We use the χ’s to cut off and compare connections and the ψ’s to cut off and compare solutions to the anti-self-dual equation on different bundles. Noting that E|_{X\x} = E_{0}|_{X\x} and setting

b := χ_{0}(A − A_{0}) ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}), (8.9)

we have A = A_{0} + b over supp ψ_{0} ⊂ X since χ_{0} = 1 on supp ψ_{0}; the section b may also be viewed as an element of L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E}).

The following comparison of solutions to the linear part of the anti-self-dual equation is the basic prototype we later use to compare solutions to the full anti-self-dual equation on different bundles 𝔤_{E} and 𝔤_{E_{0}}.

Lemma 8.3. Let A, A_{0} be connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}}, respectively, and assume that Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} = 0. Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Given w ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and w_{0} ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}), suppose v ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and v_{0} ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}) satisfy

d_{A}^{+}d_{A}^{+,∗}v = w and d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v_{0} = w_{0}.

Then,

∥ψ_{0}(v − v_{0})∥_{L^{∞}(X)} + ∥ψ_{0}(v − v_{0})∥_{L_{2,A_{0}}^{2}(X)}
≤ C∥ψ_{0}(w − w_{0})∥_{L^{♯,2}}
+ C(∥∇^{2}ψ_{0}∥_{L^{♯,2}} + ∥b∥_{L^{2♯,4}} + ∥∇_{A_{0}}b∥_{L^{♯,2}} + ∥b∥_{L^{2♯,4}}^{2}
+ ∥∇ψ_{0}∥_{L^{2♯,4}}∥b∥_{L^{2♯,4}} + ∥∇ψ_{0}∥_{L^{2♯,4}})
× (∥v∥_{L^{∞}} + ∥∇_{A}v∥_{L^{2♯,4}} + ∥v_{0}∥_{L^{∞}} + ∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}}),

HOMOTOPY EQUIVALENCE AND DONALDSON INVARIANTS. I December 14, 1998 73

where b := χ_{0}(A − A_{0}) ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}), and C(ν_{2}[A_{0}], ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}) is a constant, and ν_{2}[A_{0}] is the least positive eigenvalue of the Laplacian d_{A_{0}}^{+}d_{A_{0}}^{+,∗}.

Proof. We first compute a suitable L^{♯,2} estimate for d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}) and then apply our elliptic estimates for d_{A_{0}}^{+}d_{A_{0}}^{+,∗} given in Lemma 7.1. We have A = A_{0} + b over supp ψ_{0} ⊂ X, since χ_{0} = 1 on supp ψ_{0}, and thus on supp ψ_{0}, we see that

d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v = −d_{A_{0}}^{+} ∗ d_{A_{0}}v
= −d_{A}^{+} ∗ d_{A_{0}}v + (b ∧ ∗d_{A_{0}}v)^{+}
= −d_{A}^{+} ∗ d_{A}v + d_{A}^{+} ∗ (b ∧ v) + (b ∧ ∗d_{A}v)^{+} − (b ∧ ∗(b ∧ v))^{+},

and so d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v = d_{A}^{+}d_{A}^{+,∗}v + d_{A}^{+} ∗ (b ∧ v) + (b ∧ ∗d_{A}v)^{+} − (b ∧ ∗(b ∧ v))^{+}. (8.10)

According to our Leibnitz formula (8.8) we have

\begin{aligned}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}) & = ∇^{2}ψ_{0} ⊗ (v − v_{0}) + ∇ψ_{0} ⊗ ∇_{A_{0}}(v − v_{0}) \\ & + ψ_{0}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}(v − v_{0}). \end{aligned} (8.11)

Therefore, combining (8.10) and (8.11), we obtain

\begin{aligned}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}) & = ∇^{2}ψ_{0} ⊗ (v − v_{0}) + ∇ψ_{0} ⊗ ∇_{A_{0}}(v − v_{0}) \\ & + ψ_{0}(d_{A}^{+}d_{A}^{+,∗}v − d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v_{0}) + ψ_{0}d_{A}^{+} ∗ (b ∧ (v − v_{0})) \\ & + ψ_{0}(b ∧ ∗d_{A}(v − v_{0}))^{+} − ψ_{0}(b ∧ ∗(b ∧ (v − v_{0})))^{+}. \end{aligned} (8.12)

Consequently, as d_{A}^{+}d_{A}^{+,∗}v = w and d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v_{0} = w_{0}, we have the schematic expression, employing tensor products to denote all bilinear products and using ∇_{A_{0}}(v − v_{0}) = ∇_{A}v + b ⊗ v − ∇_{A_{0}}v_{0} in the identity (8.12),

\begin{aligned}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}) & = ψ_{0}(w − w_{0}) + ∇^{2}ψ_{0} ⊗ (v − v_{0}) \\ & + ∇ψ_{0} ⊗ (∇_{A}v − ∇_{A_{0}}v_{0} − b ⊗ v_{0})) \\ & + ψ_{0}∇_{A_{0}}b ⊗ (v − v_{0}) \\ & + ψ_{0}b ⊗ (∇_{A}v − ∇_{A_{0}}v_{0}) + ψ_{0}b ⊗ b ⊗ (v + v_{0}). \end{aligned} (8.13)

We need an L^{♯,2} bound on d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}) in terms of the L^{♯,2} norm of ψ_{0}(w − w_{0}) and the lower order terms in the expression (8.13) for d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}):

∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}}

≤ ∥ψ_{0}(w − w_{0})∥_{L^{♯,2}} + (∥∇^{2}ψ_{0}∥_{L^{♯,2}} + ∥b∥_{L^{2♯,4}} + ∥∇_{A_{0}}b∥_{L^{♯,2}} + ∥b∥_{L^{2♯,4}}^{2} (8.14)

+ ∥∇ψ_{0}∥_{L^{2♯,4}}∥b∥_{L^{2♯,4}})(∥v∥_{L^{∞}} + ∥v_{0}∥_{L^{∞}})
+ (∥∇ψ_{0}∥_{L^{2♯,4}} + ∥b∥_{L^{2♯,4}})(∥∇_{A}v∥_{L^{2♯,4}} + ∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}}).

From Lemma 7.1 and the eigenvalue bound, ∥ψ_{0}(v − v_{0})∥_{L^{2}} ≤ ν_{2}[A_{0}]^{−1}∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{2}},

74 PAUL M. N. FEEHAN AND THOMAS G. LENESS December 14, 1998

we have the elliptic estimate ∥ψ_{0}(v − v_{0})∥_{L^{∞}} + ∥ψ_{0}(v − v_{0})∥_{L_{2,A_{0}}^{2}} ≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}}, (8.15) for a constant C(ν_{2}[A_{0}], ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}). Combining the bound (8.15) with that of (8.14) then yields the desired estimate. Remark 8.4. Lemma 8.3 allows us to compare solutions v ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) to the second order, linear equation d_{A}^{+}d_{A}^{+,∗}v = w with solutions v_{0} ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}) to d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v_{0} = w_{0} when the point [A] lies in an Uhlenbeck neighborhood of a point [A_{0}, x]. Otherwise said, it allows us to compare the second-order Green’s operator for d_{A}^{+}d_{A}^{+,∗} with the Green’s operator for d_{A_{0}}^{+}d_{A_{0}}^{+,∗} as the point [A] approaches [A_{0}, x].

It is a simpler matter to bound the L_{1}^{2} norm of the difference between v and v_{0} over compact subsets of X \ x: Lemma 8.5. Continue the notation and hypotheses of of Lemma 8.3. Then,

∥ψ_{0}(v − v_{0})∥_{L_{1,A_{0}}^{2}(X)}

≤ C(∥ψ_{0}(w − w_{0})∥_{L^{4/3}} + ∥∇^{2}ψ_{0}∥_{L^{4/3}}(∥v∥_{L^{∞}} + ∥v_{0}∥_{L^{∞}}))
+ C∥∇ψ_{0}∥_{L^{2}}(∥∇_{A}v∥_{L^{4}} + ∥∇_{A_{0}}v_{0}∥_{L^{4}} + ∥b∥_{L^{4}}∥v_{0}∥_{L^{∞}})
+ C∥∇_{A_{0}}b∥_{L^{4/3}}(∥ψ_{0}(v − v_{0})∥_{L^{∞}}
+ C(∥b∥_{L^{2}}(∥∇_{A}v∥_{L^{4}} + ∥∇_{A_{0}}v_{0})∥_{L^{4}} + ∥b∥_{L^{2}}∥b∥_{L^{4}}(∥v∥_{L^{∞}} + ∥v_{0}∥_{L^{∞}})).

where b := χ_{0}(A − A_{0}) ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}), and C(ν_{2}[A_{0}], ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}) is a constant, and ν_{2}[A_{0}] is the least positive eigenvalue of the Laplacian d_{A_{0}}^{+}d_{A_{0}}^{+,∗}. Proof. According to the elliptic estimates of Lemma 6.6, we have

∥ψ_{0}(v − v_{0})∥_{L_{1,A_{0}}^{2}} ≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{4/3}}.

On the other hand, our schematic expression (8.13) for d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0}) yields

∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{4/3}} ≤ ∥ψ_{0}(w − w_{0})∥_{L^{4/3}} + ∥∇^{2}ψ_{0}∥_{L^{4/3}}(∥v∥_{L^{∞}} + ∥v_{0}∥_{L^{∞}})
+ ∥∇ψ_{0}∥_{L^{2}}(∥∇_{A}v∥_{L^{4}} + ∥∇_{A_{0}}v_{0}∥_{L^{4}} + ∥b∥_{L^{4}}∥v_{0}∥_{L^{∞}})
+ ∥∇_{A_{0}}b∥_{L^{4/3}}∥ψ_{0}(v − v_{0})∥_{L^{∞}}
+ ∥b∥_{L^{2}}(∥∇_{A}v∥_{L^{4}} + ∥∇_{A_{0}}v_{0})∥_{L^{4}}
+ ∥b∥_{L^{2}}∥b∥_{L^{4}}(∥v∥_{L^{∞}} + ∥v_{0}∥_{L^{∞}}).

Combining the preceding estimates gives the desired bound.

It remains to give a bound for the difference ψ_{0}(a − a_{0}). Lemma 8.6. Let A, A_{0} be connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}}, respectively, over X. Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Given w ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E})

HOMOTOPY EQUIVALENCE AND DONALDSON INVARIANTS. I December 14, 1998 75

and w_{0} ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}), suppose v ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and v_{0} ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}). Let a(A) := d_{A}^{+,∗}v ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E}) and a_{0} := a(A_{0}) = d_{A_{0}}^{+,∗}v_{0} ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}). Then

∥ψ_{0}(a − a_{0})∥_{L_{1,A_{0}}^{2}} ≤ ∥ψ_{0}(v − v_{0})∥_{L_{2,A_{0}}^{2}}
+ (∥b∥_{L_{1,A_{0}}^{2}} + ∥b∥_{L^{4}}∥∇ψ_{0}∥_{L^{4}} + ∥∇ψ_{0}∥_{L_{1}^{2}})(∥v_{0}∥_{L^{∞}} + ∥v∥_{L^{∞}})
+ (∥∇ψ_{0}∥_{L^{4}} + ∥b∥_{L^{4}})(∥∇_{A_{0}}v∥_{L^{4}} + ∥∇_{A}v∥_{L^{4}} + ∥b∥_{L^{4}}∥v∥_{L^{∞}}).

where b = χ_{0}(A − A_{0}) ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}). Proof. Schematically, we have

ψ_{0}(a − a_{0}) = ψ_{0}(∇_{A}v − ∇_{A_{0}}v_{0})

= ∇_{A}ψ_{0}(v − v_{0}) − ∇ψ_{0} ⊗ (v − v_{0}) (8.16)

= ∇_{A_{0}}ψ_{0}(v − v_{0}) + b ⊗ ψ_{0}(v − v_{0}) − ∇ψ_{0} ⊗ (v − v_{0}).

Thus, we can bound the L_{1,A_{0}}^{2} norm of ψ_{0}(a−a_{0}) in terms of the L_{1,A_{0}}^{2} norm of ∇_{A_{0}}(ψ_{0}(v − v_{0})) and the lower-order terms in the expression (8.16) for ψ_{0}(a − a_{0}). For the L^{2} norm we have

\begin{aligned}∥ψ_{0}(a − a_{0})∥_{L^{2}} & ≤ ∥∇_{A_{0}}ψ_{0}(v − v_{0})∥_{L^{2}} \\ & + (∥b∥_{L^{2}} + ∥∇ψ_{0}∥_{L^{2}})(∥v∥_{L^{∞}} + ∥v_{0}∥_{L^{∞}}). \end{aligned} (8.17)

To compute the L^{2} norm of ∇_{A_{0}}ψ_{0}(a − a_{0}), note that, schematically, the identity (8.16) yields

∇_{A_{0}}ψ_{0}(a − a_{0}) = ∇_{A_{0}}^{2}ψ_{0}(v − v_{0}) + ∇_{A_{0}}b ⊗ ψ_{0}(v − v_{0})
+ b ⊗ ∇ψ_{0} ⊗ (v − v_{0}) + b ⊗ ψ_{0}∇_{A_{0}}(v − v_{0})
− ∇^{2}ψ_{0} ⊗ (v − v_{0}) − ∇ψ_{0} ⊗ ∇_{A_{0}}(v − v_{0}),

with ∇_{A_{0}}v = ∇_{A}v − b ⊗ v. Therefore,

∥∇_{A_{0}}ψ_{0}(a − a_{0})∥_{L^{2}} ≤ ∥∇_{A_{0}}^{2}ψ_{0}(v − v_{0})∥_{L^{2}}

+ (∥∇_{A_{0}}b∥_{L^{2}} + ∥b∥_{L^{4}}∥∇ψ_{0}∥_{L^{4}} + ∥∇^{2}ψ_{0}∥_{L^{2}})(∥v_{0}∥_{L^{∞}} + ∥v∥_{L^{∞}})

+ (∥∇ψ_{0}∥_{L^{4}} + ∥b∥_{L^{4}})(∥∇_{A_{0}}v∥_{L^{4}} + ∥∇_{A}v∥_{L^{4}} + ∥b∥_{L^{4}}∥v∥_{L^{∞}}).

Combining this estimate with the L^{2} bound (8.17) then gives the desired result.

Again, the L^{2} bound is simpler: Lemma 8.7. Continue the notation of Lemma 8.6. Then,

∥ψ_{0}(a − a_{0})∥_{L^{2}(X)} ≤ ∥ψ_{0}(v − v_{0})∥_{L_{1,A_{0}}^{2}}
+ (∥b∥_{L^{2}} + ∥∇ψ_{0}∥_{L^{2}})∥χ_{0}(v − v_{0})∥_{L^{∞}}.

Proof. From the schematic expression (8.16) for ψ_{0}(a − a_{0}) we see that

∥ψ_{0}(a − a_{0})∥_{L^{2}} ≤ ∥∇_{A_{0}}ψ_{0}(v − v_{0})∥_{L^{2}}
+ (∥b∥_{L^{2}} + ∥∇ψ_{0}∥_{L^{2}})∥χ_{0}(v − v_{0})∥_{L^{∞}}.

This completes the proof.

It remains to bound the Sobolev norms of b with estimates for Sobolev norms of A − A_{0} and χ_{0}.

Lemma 8.8. Continue the above notation. Then the following estimates hold,

(1)  ∥b∥_{L^{2♯,4}(X)} ≤ c∥b∥_{L_{1,A_{0}}^{2}(X)},
(2)  ∥b∥_{L_{1,A_{0}}^{2}(X)} ≤ (1 + ∥∇χ_{0}∥_{L^{2}(X)})∥A − A_{0}∥_{L_{1,A_{0}}^{2}(U)},
(3)  ∥∇_{A_{0}}b∥_{L^{♯,2}(X)} ≤ C(1 + ∥∇χ_{0}∥_{L^{4}(X)})∥A − A_{0}∥_{L_{2,A_{0}}^{2}(U)},

where c depends only on the metric g, while C depends on U and g.

Proof. Assertion 1 follows immediately from [12, Lemma 4.1]. Next, using ∇_{A_{0}}χ_{0}(A−A_{0}) = ∇χ_{0} ⊗ (A − A_{0}) + χ_{0}∇_{A_{0}}(A − A_{0}), we see that

∥b∥_{L_{1,A_{0}}^{2}(X)} ≤ (1 + ∥∇χ_{0}∥_{L^{2}(X)})∥A − A_{0}∥_{L^{2}(U)} + ∥∇_{A_{0}}(A − A_{0})∥_{L^{2}(U)},

as required for Assertion 2. Lastly, for any 2 < p < 4 and 4 < q < ∞ for which 1/p = 1/4 + 1/q, the Sobolev embeddings and Hölder inequality yield

∥∇_{A_{0}}b∥_{L^{♯}(X)} ≤ c∥∇_{A_{0}}b∥_{L^{p}(X)}
≤ c∥∇χ_{0}∥_{L^{4}(X)}∥A − A_{0}∥_{L^{q}(U)} + ∥∇_{A_{0}}(A − A_{0})∥_{L^{p}(U)},
≤ C∥∇χ_{0}∥_{L^{4}(X)}∥A − A_{0}∥_{L_{2,A_{0}}^{2}(U)} + C∥∇_{A_{0}}(A − A_{0})∥_{L_{1,A_{0}}^{2}(U)},

which completes the proof of Assertion 3.

Remark 8.9. Although we assumed Ker d_{A_{0}}^{+}d_{A_{0}}^{+,∗} = 0 in Lemmas 8.3 and 8.5, we did not need to explicitly need this assumption in Lemmas 8.6 and 8.8.

8.3. Degeneration of solutions to the second-order quasi-linear equation: nonlinear L_{2}^{2} theory. We now consider the difference of the non-linear terms in the anti-self-dual equation and, in conjunction with Lemma 8.3, we obtain the following comparison of solutions to the anti-self-dual equation.

Proposition 8.10. Let A, A_{0} be connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}}, respectively, over (X, g). Then there are positive constants C = C(ν_{2}[A_{0}], ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}) and ε = ε(C), ν_{2}[A_{0}] is the least positive eigenvalue of the Laplacian d_{A_{0}}^{+}d_{A_{0}}^{+,∗}, such that the following holds. Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Given w ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and w_{0} ∈ L_{k−1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}), suppose v ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E}) and v_{0} ∈ L_{k+1}^{2}(X, Λ^{+} ⊗ 𝔤_{E_{0}}) satisfy equations (8.3) and (8.4), respectively. If

max {∥b∥_{L^{2♯,4}}, ∥∇ψ_{0}∥_{L^{2♯,4}}, ∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}}, ∥∇_{A}v∥_{L^{2♯,4}}} < ε, (8.18) where b = χ_{0}(A − A_{0}) ∈ L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E_{0}}), then

\begin{matrix}∥ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}(X)} \\ ≤ C(∥ψ_{0}(w − w_{0})∥_{L^{♯,2}} + ∥b∥_{L^{2♯,4}}∥v∥_{L^{∞}}∥∇_{A}v∥_{L^{2♯,4}} + ∥b∥_{L^{2♯,4}}^{2}∥v∥_{L^{∞}}^{2}) \\ + C∥∇_{A_{0}}ψ_{0}(v − v_{0})∥_{L^{2♯,4}}(∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}} + ∥∇_{A_{0}}v∥_{L^{2♯,4}} + ∥b∥_{L^{2♯,4}}∥v∥_{L^{∞}}) \\ + (∥∇^{2}ψ_{0}∥_{L^{♯,2}} + ∥∇ψ_{0}∥_{L^{4}(X)} + ∥b∥_{L^{4}} + ∥∇_{A_{0}}b∥_{L^{♯,2}}) \\ × (∥v∥_{L^{∞}∩L_{2,A}^{2}} + ∥v_{0}∥_{L^{∞}∩L_{2,A_{0}}^{2}} ). \end{matrix} (8.19)

Proof. From equations (8.3) and (8.4) we have

\begin{aligned}d_{A}^{+}d_{A}^{+,∗}v & = −(d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} + w =: w̃, \\ d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v_{0} & = −(d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v_{0})^{+} + w_{0} =: w̃_{0}. \end{aligned} (8.20)

Therefore, applying Lemma 8.3 and the embedding L_{1}^{2} ⊂ L^{2♯,4} [12, Lemma 4.1] to the pair of equations (8.20) yields the estimate (when ∥b∥_{L_{1,A_{0}}^{2}} ≤ 1)

\begin{aligned}∥ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} & ≤ C∥ψ_{0}(w̃ − w̃_{0})∥_{L^{♯,2}} \\ & + C (∥∇^{2}ψ_{0}∥_{L^{♯,2}} + ∥∇ψ_{0}∥_{L^{4}} + ∥b∥_{L^{4}} + ∥∇_{A_{0}}b∥_{L^{♯,2}}) \\ & × (∥v∥_{L^{∞}∩L_{2,A}^{2}} + ∥v_{0}∥_{L^{∞}∩L_{2,A_{0}}^{2}} ), \end{aligned} (8.21)

where b and the constant C are as in the statement of the proposition; this gives the first and last terms on the right of inequality (8.19).

It remains to bound the term ψ_{0}(w̃ − w̃_{0}) (giving the remaining terms in the bound (8.19)): this difference may be written schematically as

ψ_{0}(w̃ − w̃_{0}) = ψ_{0}(w − w_{0}) + ψ_{0} (∇_{A}v ⊗ ∇_{A}v − ∇_{A_{0}}v_{0} ⊗ ∇_{A_{0}}v_{0}) . (8.22)

On supp ψ_{0} we have

(8.23)  ∇_{A}v ⊗ ∇_{A}v = ∇_{A_{0}}v ⊗ ∇_{A_{0}}v + b ⊗ v ⊗ ∇_{A_{0}}v
+ ∇_{A_{0}}v ⊗ b ⊗ v + b ⊗ v ⊗ b ⊗ v.

Now, using v = (v − v_{0}) + v_{0} on supp ψ_{0} we observe that

(8.24)  ∇_{A_{0}}v ⊗ ∇_{A_{0}}v = ∇_{A_{0}}(v − v_{0}) ⊗ ∇_{A_{0}}(v − v_{0}) + ∇_{A_{0}}v_{0} ⊗ ∇_{A_{0}}(v − v_{0})
+ ∇_{A_{0}}(v − v_{0}) ⊗ ∇_{A_{0}}v_{0} + ∇_{A_{0}}v_{0} ⊗ ∇_{A_{0}}v_{0}.

Lastly, note that

ψ_{0}∇_{A_{0}}(v − v_{0}) ⊗ ∇_{A_{0}}v_{0} = ∇_{A_{0}}ψ_{0}(v − v_{0}) ⊗ ∇_{A_{0}}v_{0}
− ∇ψ_{0} ⊗ (v − v_{0}) ⊗ ∇_{A_{0}}v_{0}
(8.25)  ψ_{0}∇_{A_{0}}v_{0} ⊗ ∇_{A_{0}}(v − v_{0}) = ∇_{A_{0}}v_{0} ⊗ ∇_{A_{0}}ψ_{0}(v − v_{0})
− ∇_{A_{0}}v_{0} ⊗ ∇ψ_{0} ⊗ (v − v_{0})
ψ_{0}∇_{A_{0}}(v − v_{0}) ⊗ ∇_{A_{0}}(v − v_{0}) = ∇_{A_{0}}ψ_{0}(v − v_{0}) ⊗ ∇_{A_{0}}(v − v_{0})
− ∇ψ_{0} ⊗ (v − v_{0}) ⊗ ∇_{A_{0}}(v − v_{0}).

Combining equations (8.22), (8.23), (8.24), and (8.25), we finally arrive at a schematic equation for the difference:

ψ_{0}(w̃ − w̃_{0})
= ψ_{0}(w − w_{0})

+ ψ_{0}(b ⊗ v ⊗ ∇_{A_{0}}v + ∇_{A_{0}}v ⊗ b ⊗ v + b ⊗ v ⊗ b ⊗ v) (8.26)

+ ∇_{A_{0}}ψ_{0}(v − v_{0}) ⊗ ∇_{A_{0}}v_{0} − ∇ψ_{0} ⊗ (v − v_{0}) ⊗ ∇_{A_{0}}v_{0}
+ ∇_{A_{0}}v_{0} ⊗ ∇_{A_{0}}ψ_{0}(v − v_{0}) − ∇_{A_{0}}v_{0} ⊗ ∇ψ_{0} ⊗ (v − v_{0})
+ ∇_{A_{0}}ψ_{0}(v − v_{0}) ⊗ ∇_{A_{0}}(v − v_{0}) − ∇ψ_{0} ⊗ (v − v_{0}) ⊗ ∇_{A_{0}}(v − v_{0}).

Hence, from equation (8.26), Hölder’s inequality, and the Sobolev multiplication L^{2♯,4} ⊗ L^{2♯,4} → L^{♯,2} [12, Lemma 4.3], we have a bound

∥ψ_{0}(w̃ − w̃_{0})∥_{L^{♯,2}}

≤ C (∥ψ_{0}(w − w_{0})∥_{L^{♯,2}} + ∥b∥_{L^{2♯,4}}∥v∥_{L^{∞}}∥∇_{A}v∥_{L^{2♯,4}} + ∥b∥_{L^{2♯,4}}^{2}∥v∥_{L^{∞}}^{2} (8.27)

+ ∥∇_{A_{0}}ψ_{0}(v − v_{0})∥_{L^{2♯,4}} (∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}} + ∥∇_{A_{0}}v∥_{L^{2♯,4}} + ∥b∥_{L^{2♯,4}}∥v∥_{L^{∞}})
+∥∇ψ_{0}∥_{L^{2♯,4}}∥v − v_{0}∥_{L^{∞}} (∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}} + ∥∇_{A}v∥_{L^{2♯,4}} + ∥b∥_{L^{2♯,4}}∥v∥_{L^{∞}})) .

For small enough ε, our hypotheses (8.18) implies that the terms ∥b∥_{L^{2♯,4}}, ∥∇ψ_{0}∥_{L^{2♯,4}}, ∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}}, and ∥∇_{A}v∥_{L^{2♯,4}} are small enough to allow us to rearrange the bound (8.21) and, together with the estimate (8.27), yield the desired bound (8.19).

Remark 8.11. Just as in §8.2, Lemma 8.6 provides an L_{1,A_{0}}^{2} bound for the difference ψ_{0}(a − a_{0}), while Lemma 8.8 gives a bound for the Sobolev norms of b in terms of estimates on the Sobolev norms of A − A_{0} and χ_{0}.

Remark 8.12. To motivate the last paragraph of the proof of Proposition 8.10, we note that in our application A will be close to A_{0} in L_{k,A_{0},loc}^{2} on X \ x, so the term ∥b∥_{L^{2♯,4}} will be close to zero. We can also choose ψ_{0} such that ∥∇ψ_{0}∥_{L^{2♯,4}} is small. Since v and v_{0} solve the second-order anti-self-dual equations, we may assume that the terms ∥∇_{A_{0}}v_{0}∥_{L^{2♯,4}} and ∥∇_{A}v∥_{L^{2♯,4}} are small, though not necessarily converging to zero if A converges to A_{0} on on X \ x.

8.4. Degeneration of solutions to the second-order extended anti-self-dual equation: L_{2}^{2} theory. The new difficulty here is that we need to estimate ∥ψ_{0}(w − w_{0})∥_{L^{♯,2}(X)} where w, w_{0} are now explicitly defined by equation (8.5), so

\begin{aligned}w(A) & := Π_{A,μ}((d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} + F_{A}^{+}) − F_{A}^{+}, \\ w_{0} & := w(A_{0}) = Π_{A_{0},μ}((d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v_{0})^{+} + F_{A_{0}}^{+}) − F_{A_{0}}^{+}, \end{aligned} (8.28)

and where v and v_{0} must satisfy Π_{A_{0},μ}v_{0} = 0 and Π_{A,μ}v = 0. Typically, we choose μ = \frac{1}{2}ν_{2}[A_{0}].

The obvious problem is that our resulting quasi-linear equation (8.3) is no longer local, that is, given purely by a partial differential equation, as we have included spectral projections. However, by virtue of our comparison in §6 of the eigenvectors of d_{A}^{+}d_{A}^{+,∗} with the approximate eigenvectors defined by cutting off those of d_{A_{0}}^{+}d_{A_{0}}^{+,∗}, we can replace our new non-local quasi-linear equations with ‘essentially local’ ones plus error terms which converge to zero if A converges to (A_{0}, x) in the Uhlenbeck sense. Given this broad outline, we proceed to the details.

Our task is to compare the solution v to equation (8.3) with the solution v_{0} to the corresponding equation (8.4), where w and w_{0} are defined by (8.28). Hence, the proof proceeds in much the same way as that of Proposition 8.10, the main additional step being that in order to estimate the L^{♯,2} norms of the cut-off differences, ψ_{0}(w − w_{0}), we need to also compare the L^{2}-orthogonal projections Π_{A_{0},μ} and Π_{A,μ}.

8.4.1. Preliminary estimates. With the aid of two elementary lemmas and our eigenvector estimates from §6, we can effectively reduce the problem of estimating the difference between solutions to the extended anti-self-dual equation to the simpler case of the ordinary antiself-dual equation.

We begin with the more specific problem of estimating v − v_{0} on the subset {x ∈ X : ψ_{0}(x) = 1}.

Lemma 8.13. Let X be a closed, oriented, smooth, Riemannian four-manifold. Let A, A_{0} be L_{4}^{2} connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}} over X, respectively. Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Let v ∈ L_{3}^{2}(Λ^{+} ⊗𝔤_{E}), v_{0} ∈ L_{3}^{2}(Λ^{+} ⊗𝔤_{E_{0}}), and suppose that Π_{A,μ}v = 0 and Π_{A_{0},μ}v_{0} = 0. Then,

∥ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}(X)} ≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}(X)}
+ C(∥1 − ψ_{0}∥_{L^{2}(X)}(∥v_{0}∥_{L^{∞}(X)} + ∥v∥_{L^{∞}(X)})
+ ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}v∥_{L^{2}(X)}).

where C = C(μ, ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}).

Proof. Our estimate for ∥ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} is initiated by

\begin{aligned}∥ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} & ≤ ∥Π_{A_{0},μ}^{⊥}ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} \\ & + ∥Π_{A_{0},μ}ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} . \end{aligned} (8.29)

The last term on the right in (8.29) is ‘small’ because Π_{A,μ}v = 0 and Π_{A_{0},μ}v_{0} = 0. Indeed,

∥Π_{A_{0},μ}ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} ≤ C∥Π_{A_{0},μ}ψ_{0}(v − v_{0})∥_{L^{2}} (by Lemma 6.7)
≤ C(∥Π_{A_{0},μ}v_{0}∥_{L^{2}} + ∥Π_{A_{0},μ}(1 − ψ_{0})v_{0}∥_{L^{2}}
+ ∥Π_{A,μ}ψ_{0}v∥_{L^{2}} + ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}v∥_{L^{2}})
≤ C(∥Π_{A_{0},μ}(1 − ψ_{0})v_{0}∥_{L^{2}} + ∥Π_{A,μ}v∥_{L^{2}}
+ ∥Π_{A,μ}(1 − ψ_{0})v∥_{L^{2}} + ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}v∥_{L^{2}}),

and thus

\begin{aligned}∥Π_{A_{0},μ}ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}} & ≤ C(∥1 − ψ_{0}∥_{L^{2}}(∥v_{0}∥_{L^{∞}} + ∥v∥_{L^{∞}}) \\ & + ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}v∥_{L^{2}}). \end{aligned} (8.30)

We estimate the second-last term on the right in (8.29) by

∥Π_{A_{0},μ}^{⊥}ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}}
= ∥G_{A_{0},μ}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}}
≤ C∥Π_{A_{0},μ}^{⊥}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}} by Lemma 7.1
≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}} + C∥Π_{A_{0},μ}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}}
≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}} + C∥Π_{A_{0},μ}d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{2}}
≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}} + Cμ∥Π_{A_{0},μ}ψ_{0}(v − v_{0})∥_{L^{2}}.

(8.31)

The last term on the right in (8.31) can be estimated via inequality (8.30). The conclusion follows by combining inequalities (8.29), (8.30), and (8.31).

Suppose u ∈ Γ(Λ^{+} ⊗ 𝔤_{E}) and u_{0} ∈ Γ(Λ^{+} ⊗ 𝔤_{E_{0}}). From equations (8.3), (8.4), (8.28) and Proposition 8.10, we see that we shall need at least L^{♯,2}(X) estimates for terms of the general form ψ_{0}(Π_{A,μ}u − Π_{A_{0},μ}u_{0}).

Lemma 8.14. Let X be a closed, oriented, smooth, Riemannian four-manifold. Let A, A_{0} be L_{4}^{2} connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}} over X, respectively. Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Continue the notation of the preceding paragraph. Then, for any 1 ≤ ℓ ≤ k,

∥ψ_{0}(Π_{A,μ}u − Π_{A_{0},μ}u_{0})∥_{L^{♯,2}(X)} ≤ C(∥ψ_{0}(u − u_{0})∥_{L^{2}(X)} + ∥Π_{A,μ}(1 − ψ_{0})u∥_{L^{2}(X)}
+ ∥Π_{A_{0},μ}(1 − ψ_{0})u_{0}∥_{L^{2}(X)}
+ ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}u∥_{L^{2}(X)}),

where C = C(μ, ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}).

Proof. Observe that, via the embedding L^{4}(X) ⊂ L^{♯}(X) of [12, Lemma 4.1] and L_{1,A_{0}}^{2}(X) ⊂ L^{4}(X) of Lemma 6.5,

∥ψ_{0}(Π_{A,μ}u − Π_{A_{0},μ}u_{0})∥_{L^{♯,2}} ≤ ∥Π_{A,μ}u − Π_{A_{0},μ}u_{0}∥_{L^{♯,2}}
≤ C∥Π_{A,μ}u − Π_{A_{0},μ}u_{0}∥_{L^{2}} (by Lemma 6.7, above embeddings)
≤ C(∥Π_{A,μ}ψ_{0}u − Π_{A_{0},μ}ψ_{0}u_{0}∥_{L^{2}} + ∥Π_{A,μ}(1 − ψ_{0})u∥_{L^{2}}
+ ∥Π_{A_{0},μ}(1 − ψ_{0})u_{0}∥_{L^{2}})
≤ C(∥Π_{A_{0},μ}ψ_{0}(u − u_{0})∥_{L^{2}} + ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}u∥_{L^{2}}
+ ∥Π_{A,μ}(1 − ψ_{0})u∥_{L^{2}} + ∥Π_{A_{0},μ}(1 − ψ_{0})u_{0}∥_{L^{2}}),

and the conclusion follows.

Remark 8.15. Recall from §6 that we can approximate Π_{A,μ} by L^{2}-orthogonal projection onto a space of vectors in Γ(Λ^{+} ⊗ 𝔤_{E}) with support contained in {x ∈ X : ψ_{0}(x) = 1}; similarly for Π_{A_{0},μ}.

With the aid of Proposition 8.10 and its proof, we obtain the following comparison result for solutions to the extended anti-self-dual equations.

Proposition 8.16. Let A, A_{0} be L_{4}^{2} connections on SO(3) bundles 𝔤_{E}, 𝔤_{E_{0}}, respectively, over X. Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Suppose that v ∈ L_{3}^{2}(X, Λ^{+}⊗𝔤_{E}) and v_{0} ∈ L_{3}^{2}(X, Λ^{+}⊗𝔤_{E_{0}}) satisfy equations (8.3) and (8.4), respectively, with w, w_{0} defined by equations (8.28). Then

∥ψ_{0}(v − v_{0})∥_{L^{∞}(X)} + ∥ψ_{0}(v − v_{0})∥_{L_{2,A_{0}}^{2}(X)}

≤ C∥ψ_{0}(F_{A}^{+} − F_{A_{0}}^{+})∥_{L^{♯,2}} + C(∥1 − ψ_{0}∥_{L^{2}(X)}(∥v_{0}∥_{L^{∞}(X)} + ∥v∥_{L^{∞}(X)}) + ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}v∥_{L^{2}(X)}) + C(∥Π_{A,μ}(1 − ψ_{0})u∥_{L^{2}(X)} + ∥Π_{A_{0},μ}(1 − ψ_{0})u_{0}∥_{L^{2}(X)} + ∥(Π_{A,μ} − Π_{A_{0},μ})ψ_{0}u∥_{L^{2}(X)}) + All terms on right of inequality (8.19) in Proposition 8.10, (8.32)

where u, u_{0} are defined by (8.33), b = χ_{0}(A−A_{0}) ∈ L_{4}^{2}(X, Λ^{1}⊗𝔤_{E_{0}}) and C = C(μ, ∥F_{A_{0}}∥_{L^{2}}, ∥F_{A_{0}}^{+}∥_{L^{∞}}) is a constant.

Proof. According to Lemma 8.13, we have

∥ψ_{0}(v − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}(X)} ≤ C∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}(X)} + Small terms,

where the ‘small terms’ in question (those which tend to zero if A converges to (A_{0}, x) in the Uhlenbeck topology) have been added to the right-hand side of (8.32) above. To estimate the term ∥d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{0}(v − v_{0})∥_{L^{♯,2}(X)}, we apply Proposition 8.10 to reduce the problem to one of estimating the term ∥ψ_{0}(w − w_{0})∥_{L^{♯,2}}, with w, w_{0} defined by equations (8.28), again modulo small terms which have been added to the right-hand side of inequality (8.32).

To bound the term ∥ψ_{0}(w − w_{0})∥_{L^{♯,2}}, we apply Lemma 8.14, where we choose

\begin{aligned}u(A) & := (d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v)^{+} + F_{A}^{+}, \\ u_{0} & := u(A_{0}) = (d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v_{0})^{+} + F_{A_{0}}^{+}. \end{aligned} (8.33)

The other term we need to estimate is ∥ψ_{0}(F_{A}^{+} − F_{A_{0}}^{+})∥_{L^{♯,2}(X)}, which requires no further comment. From Lemma 8.14 we see that it suffices to estimate the term ∥ψ_{0}(u − u_{0})∥_{L^{2}}, as again the remaining terms from Lemma 8.14 are small and have been added to the right-hand side of inequality (8.32). Using the definition of u, u_{0} in (8.33) we see from the proof of Proposition 8.10 that, starting from the definition of w̃, w̃_{0} in (8.20) and using rearrangement, the term ∥ψ_{0}(u − u_{0})∥_{L^{2}} is bounded by terms already appearing on the right-hand side of inequality (8.32).

Proposition 8.16, in conjunction with the supplementary lemma below, shows that if {A_{α}} converges to (A_{0}, x) in the Uhlenbeck sense, then indeed v_{α} = v(A_{α}) converges to v_{0}(A_{0}) in L_{2,A_{0},loc}^{2}(X \ x) and w_{α} = w(A_{α}) converges to w_{0}(A_{0}) in L_{loc}^{♯,2}(X \ x).

Lemma 8.17. Let the hypotheses be as in Theorem 8.2. Then the following hold:

• The sequence of solutions {v_{α}} to equation (8.1) converges to the solution v_{0} to equation (8.2) in L^{∞} ∩ L_{2,A_{0},loc}^{2} on X \ x,

• The sequence of terms {w_{α}} converges to the term w_{0} in L_{loc}^{♯,2} on X \ x, • The sequence of solutions a_{α} = d_{A_{α}}^{+,∗}v_{α} to equation (8.1) converges to the solution

a_{0} = d_{A_{0}}^{+,∗}v_{0} to (8.2) in L_{1,A_{0},loc}^{2} on X \ x,

• The sequence of connections {A_{α} + a_{α}} converges to A_{0} + a_{0} in L_{1,A_{0},loc}^{2} on X \ x.

Proof. We first consider the terms contributed to the right-hand side of inequality (8.32) in Proposition 8.16 by Proposition 8.10. Let C be the constant of Proposition 8.10. Suppose U ⋐ X \ x is a precompact open subset and that ε is a positive constant. By Lemma 6.9 we can choose a sequence of cutoff functions {ψ_{n}} on X which are x-good in the sense that

• ψ_{n} = 1 on U,

• supp ψ_{n} ⊂ V_{n} ⋐ X\x, where {V_{n}} is a collection of open subsets such that ∪V_{n} = X\x,

• ψ_{n} converges pointwise to the constant 1 on X \ x, and

• lim_{n→∞} ∥∇ψ_{n}∥_{L^{2♯,4}(X)} + ∥∇^{2}ψ_{n}∥_{L^{♯,2}(X)} = 0. We shall use the cutoff functions ψ_{n} in place of the ψ_{0} appearing in the statement of Proposition 8.10. In particular, we may suppose that for large enough n_{0}(ε, C, M) (letting M be the constant in the inequalities (8.37)),

∥∇^{2}ψ_{n}∥_{L^{♯,2}(X)} + ∥∇ψ_{n}∥_{L^{2♯,4}(X)} ≤ ε/(1000CM^{2}), n ≥ n_{0}. (8.34)

For any fixed n, the sequence {A_{α}} converges to A_{0} in L_{k,A_{0}}^{2}(V_{n}) and so, for large enough α_{0} = α_{0}(C, ε, n), we have

∥A_{α} − A_{0}∥_{L_{k,A_{0}}^{2}(V_{n})} ≤ ε/(1000CM^{2}), α ≥ α_{0}. (8.35)

For the remainder of the section we employ the abbreviation

b_{α} := A_{α} − A_{0} on X \ x. (8.36)

Using F (A_{α}) = F (A_{0} + b_{α}) = F_{A_{0}} + d_{A_{0}}b_{α} + b_{α} ∧ b_{α} on X \ x and the Sobolev embeddings of [12, Lemma 4.1], we see that

∥ψ_{n}(F_{A_{α}}^{+} − F_{A_{0}}^{+})∥_{L^{♯,2}(X)} ≤ ∥∇_{A_{0}}b_{α}∥_{L^{♯,2}(V_{n})} + ∥b_{α}∥_{L^{2♯,4}(V_{n})} ≤ c∥b_{α}∥_{L_{2,A_{0}}^{2}(V_{n})}
≤ c∥A_{α} − A_{0}∥_{L_{2,A_{0}}^{2}(V_{n})} ≤ ε/(1000CM^{2}).

We note that Lemma 7.4 (with right-hand side data −Π_{A_{α},μ}^{⊥}F_{A_{α}}^{+}) provides uniform bounds M on v_{α} which only depend on sup_{α}{∥F_{A_{α}}^{+}∥_{L^{∞}}, ∥F_{A_{α}}∥_{L^{2}}}, μ, g, and similarly for v_{0}, so

∥v_{α}∥_{L^{∞}∩L_{2,Aα}^{2}(X)} ≤ M and ∥v_{0}∥_{L^{∞}∩L_{2,A_{0}}^{2}(X)} ≤ M. (8.37)

Since U ⊂ {ψ_{n} = 1}, by definition of ψ_{n}, we have

∥v_{α} − v_{0}∥_{L^{∞}∩L_{2,A_{0}}^{2}(U)} ≤ ∥ψ_{n}(v_{α} − v_{0})∥_{L^{∞}∩L_{2,A_{0}}^{2}(X)}.

Hence, for large enough n_{0}, the sum of the terms contributed to the right-hand side of inequality (8.32) in Proposition 8.16 by (8.19), is bounded by ε/3, for all α ≥ α_{0}.

We now consider the terms on the right-hand side of inequality (8.32) which are contributed by Lemma 8.13. From the definition of the sequence of cutoff functions {ψ_{n}}, we have lim_{n→∞} ∥1−ψ_{n}∥_{L^{2}(X)} = 0. By §6 we know that ∥Π_{A_{α},μ}ψ_{n}−Π_{A_{0},μ}ψ_{n}∥ converges to zero as α → ∞ (for any fixed n), where ∥ · ∥ denotes the operator norm in End(L^{2}(Λ^{+} ⊗ 𝔤_{E})). Therefore, for large enough n_{0}, the sum of the terms contributed to the right-hand side of inequality (8.32) in Proposition 8.16 by Lemma 8.13, may be bounded by ε/3 for all α ≥ α_{0}. (The term d_{A_{0}}^{+}d_{A_{0}}^{+,∗}ψ_{n}(v_{α} − v_{0}) is excluded, as it has already been accounted for by Proposition 8.10.)

Lastly, we consider the terms in inequality (8.32) contributed by Lemma 8.14. From the definition of u_{α} = u(A_{α}) ∈ L^{♯,2}(Λ^{+} ⊗ 𝔤_{E}) in (8.33) and our L^{∞} ∩ L_{2,A_{α}}^{2} estimates for v_{α} in (8.37), we have a uniform L^{2} bound on u_{α}. Hence, we can see that the term ∥Π_{A_{α},μ}(1 − ψ_{n})u_{α}∥_{L^{2}(X)} converges to zero either by (i) approximating Π_{A_{α},μ} by L^{2}-orthogonal projection onto a subspace of L^{2}(Λ^{+} ⊗ 𝔤_{E}) spanned by approximate eigenvectors of d_{A_{α}}^{+}d_{A_{α}}^{+,∗} which are supported in {x ∈ X : 1 − ψ_{n}(x) = 0}, or (ii) applying the Lebesgue dominated convergence theorem to the sequence (1 − ψ_{n})u_{α}, noting that the support of 1 − ψ_{n} shrinks. The same remarks apply to the term ∥Π_{A_{0},μ}(1 − ψ_{n})u_{0}∥_{L^{2}(X)}. As with the corresponding term involving v_{α} in Lemma 8.13, the term ∥(Π_{A_{α},μ} − Π_{A_{0},μ})ψ_{n}u_{α}∥_{L^{2}(X)} also converges to zero as α → ∞, for any fixed n. Thus, the sum of the terms contributed to the right-hand side of inequality (8.32) in Proposition 8.10 by Lemma 8.14, may be bounded by ε/3, as

(8.38)  lim_{n→∞} lim_{α→∞} ∥ψ_{n}(Π_{A_{α},μ}u_{α} − Π_{A_{0},μ}u_{0})∥_{L^{2}(X)} = 0.

Consequently, summing all three contributions to the right-hand side of inequality (8.32), and taking first n, then α sufficiently large yields

∥v_{α} − v_{0}∥_{L^{∞}∩L_{2,A_{0}}^{2}(U)} ≤ ε.

Hence, the sequence {v_{α}} converges to v_{0} in L^{∞} ∩ L_{2,A_{0}}^{2}(U). Lemma 8.6 then implies that {a_{α}} converges to a_{0} in L_{1,A_{0}}^{2}(U) and thus the sequence of connections {A_{α} + a_{α}} converges to A_{0} in L_{1,A_{0}}^{2}(U).

8.5. Uhlenbeck convergence of solutions to the anti-self-dual equation: L_{k,loc}^{2} convergence. The estimates of the preceding subsection imply that ψ_{n}a_{α} converges to a_{0} in L_{1,A_{0}}^{2}(X) if the sequence of connections A_{α} converges to (A_{0}, x) in L_{k,A_{0},loc}^{2}(X \ x). It remains to show, with the aid of our higher-order interior elliptic estimates, that (i) the sequence of one-forms a_{α} converges to a_{0} in L_{ℓ,A_{0},loc}^{2} over X \ x for all 2 ≤ ℓ ≤ k and that (ii) if the sequence of measures |F_{A_{α}}|^{2} converges to |F_{A_{0}}|^{2} + 8π^{2} ∑_{x∈x} δ_{x} in the weak-* topology on measures, then the sequence of measures |F_{A_{α}+a_{α}}|^{2} also converges to |F_{A_{0}}|^{2} +8π^{2} ∑_{x∈x} δ_{x}. Note that given A_{α} converges to (A_{0}, x), we are not at liberty to pass to a subsequence in order to show that A_{α} + a_{α} converges to (A_{0}, x).

Lemma 8.18. Suppose v, v_{0} are solutions to equations (8.3) and (8.4), respectively, defined by connections A on 𝔤_{E} and A_{0} on 𝔤_{E_{0}}, with w and w_{0} defined by (8.28). Suppose U ⊂ X is an open subset such that E|_{U} = E_{0}|_{U}. Then v′ := v − v_{0} solves

d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v′ + (d_{A_{0}}^{+,∗}v′ ∧ d_{A_{0}}^{+,∗}v′)^{+} + {d_{A_{0}}^{+,∗}v_{0}, d_{A_{0}}^{+,∗}v′} (8.39)

= w′ =: w − w_{0} + b ⊗ v ⊗ d_{A_{0}}^{+,∗}v + b ⊗ b ⊗ v ⊗ v,

where b := (A − A_{0})|_{U} and {α, β} := (α ∧ β + β ∧ α)^{+}.

Proof. We work over the open set U where E = E_{0}. Using d_{A}^{+,∗}v = − ∗ d_{A}v and d_{A}^{+,∗}v = − ∗ d_{A_{0}}v − ∗(b ∧ v), we observe that

\begin{aligned}d_{A}^{+,∗}v ∧ d_{A}^{+,∗}v & = d_{A_{0}}^{+,∗}v ∧ d_{A_{0}}^{+,∗}v − ∗(b ∧ v) ∧ d_{A_{0}}^{+,∗}v − d_{A_{0}}^{+,∗}v ∧ ∗(b ∧ v) \\ & + ∗(b ∧ v) ∧ ∗(b ∧ v). \end{aligned} (8.40)

By equation (8.10) we see that

d_{A}^{+}d_{A}^{+,∗}v = d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v − (d_{A_{0}}^{+} ∗ (b ∧ v)^{+} − (b ∧ ∗d_{A_{0}}v)^{+} + (b ∧ ∗(b ∧ v))^{+}. (8.41)

Noting that v′ = v − v_{0}, we obtain the following difference between quadratic terms,

\begin{matrix}d_{A_{0}}^{+,∗}v ∧ d_{A_{0}}^{+,∗}v − d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v_{0} \\ = d_{A_{0}}^{+,∗}v′ ∧ d_{A_{0}}^{+,∗}v′ + d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v′ + d_{A_{0}}^{+,∗}v′ ∧ d_{A_{0}}^{+,∗}v_{0}. \end{matrix} (8.42)

Subtracting equation (8.4) from equation (8.3) and using the identities (8.40), (8.41), and (8.42), we obtain

d_{A_{0}}^{+}d_{A_{0}}^{+,∗}v′ + (d_{A_{0}}^{+,∗}v′ ∧ d_{A_{0}}^{+,∗}v′)^{+} + (d_{A_{0}}^{+,∗}v_{0} ∧ d_{A_{0}}^{+,∗}v′)^{+} + (d_{A_{0}}^{+,∗}v′ ∧ d_{A_{0}}^{+,∗}v_{0})^{+}
= w − w_{0} + d_{A_{0}}^{+} ∗ (b ∧ v) + (b ∧ ∗d_{A_{0}}v)^{+} − (b ∧ ∗(b ∧ v))^{+}.

This gives the desired result.

We can now complete the proofs of our main gluing-map continuity results.

Proof of Theorem 8.1. The first step is to show that the sequence of connections A_{α} + a_{α} converges to A_{0} + a_{0} in L_{ℓ,A_{0},loc}^{2}(X \ x), when 2 ≤ ℓ ≤ k; Lemma 8.17 addresses the case ℓ = 1.

Let U′′ ⋐ U′ ⋐ U ⋐ X \ x be precompact open subsets: we will show that A_{α} + a_{α} converges to A_{0} +a_{0} in L_{ℓ,A_{0}}^{2}(U′′). Observe that v′_{α} := v_{α} −v_{0} is a solution to equation (4.3) with A replaced by A_{0}, α replaced by d_{A_{0}}^{+,∗}v_{0}, and w replaced by the right-hand side, say w′_{α}, of identity (8.39). Then applying Corollary 4.14 to equation (4.3), with the preceding replacements, provides the estimate

∥v_{α} − v_{0}∥_{L_{ℓ+1,A_{0}}^{2}(U′′)} ≤ Q_{ℓ}(∥w′_{α}∥_{L_{ℓ−1,A_{0}}^{2}(U′)}, ∥v_{α} − v_{0}∥_{L^{2}(U′)}), ℓ ≥ 2. (8.43)

The positive, real coefficients of the universal polynomial Q_{ℓ} depend at most on A_{0}, v_{0}, ℓ, U′, U′′, and the metric g on X; also, Q(0, 0) = 0. We have already seen, from Lemma 8.17, that ∥v_{α} − v_{0}∥_{L^{2}(U)} converges to zero. From equations (8.28) and (8.39),

\begin{matrix}∥w′_{α}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} \\ ≤ ∥b_{α} ⊗ v_{α} ⊗ d_{A_{0}}^{+,∗}v_{α}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} + ∥b_{α} ⊗ b_{α} ⊗ v_{α} ⊗ v_{α}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} \\ + ∥Π_{A_{α},μ}u_{α} − Π_{A_{0},μ}u_{0}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} + ∥F_{A_{α}}^{+} − F_{A_{0}}^{+}∥_{L_{ℓ−1,A_{0}}^{2}(U′)}, \end{matrix} (8.44)

where u_{α}, u_{0} are defined by equation (8.33) and b_{α} by (8.36). The Sobolev multiplication theorems and definition (8.36) imply that

\begin{matrix}∥b_{α} ⊗ v_{α} ⊗ d_{A_{0}}^{+,∗}v_{α}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} + ∥b_{α} ⊗ b_{α} ⊗ v_{α} ⊗ v_{α}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} \\ ≤ \begin{cases}C∥A_{α} − A_{0}∥_{L_{3}^{2}(U′)}∥v_{α}∥_{L^{∞}∩L_{2,A_{0}}^{2}(U′)}^{2}, & ℓ = 2, \\ C∥A_{α} − A_{0}∥_{L_{ℓ}^{2}(U′)}∥v_{α}∥_{L_{ℓ,A_{0}}^{2}(U′)}^{2}, & ℓ ≥ 3, \end{cases} \end{matrix} (8.45)

for large enough α (to bound quadratic terms in ∥A_{α} − A_{0}∥_{L_{ℓ}^{2}(U′)} by ∥A_{α} − A_{0}∥_{L_{ℓ}^{2}(U′)}) while

∥F_{A_{α}}^{+} − F_{A_{0}}^{+}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} ≤ C∥A_{α} − A_{0}∥_{L_{ℓ}^{2}(U′)}. (8.46)

Using Lemma 6.9, choose a sequence of x-good cutoff functions ψ_{n} on X, so • supp ψ_{n} ⋐ X \ x,

• ψ_{n} = 1 on U, and

• ψ_{n} converges pointwise to the constant 1 on X \ x. We now exploit the fact that, for all α sufficiently large, the sequence A_{α} is uniformly L_{k,A_{0}}^{2}(U) close to A_{0} over the precompact open subset U: with this observation, our standard estimates in Lemma 6.7 for eigenvectors of d_{A_{α}}^{+}d_{A_{α}}^{+,∗} and d_{A_{0}}^{+}d_{A_{0}}^{+,∗} yield the first inequality below (the second following by definition of the ψ_{n}),

\begin{aligned}∥Π_{A_{α},μ}u_{α} − Π_{A_{0},μ}u_{0}∥_{L_{ℓ−1,A_{0}}^{2}(U′)} & ≤ C∥Π_{A_{α},μ}u_{α} − Π_{A_{0},μ}u_{0}∥_{L^{2}(U)} \\ & ≤ C∥ψ_{n}(Π_{A_{α},μ}u_{α} − Π_{A_{0},μ}u_{0})∥_{L^{2}(X)}, \end{aligned} (8.47)

where u_{α} = u(A_{α}), u_{0} = u(A_{0}) are defined by (8.33), and C = C([A_{0}], μ, U, U′, ℓ, α_{0}, g). Combining estimates (8.43), (8.44), (8.45), (8.46), and (8.47) yields the bound, for ℓ ≥ 2,

\begin{matrix}∥v_{α} − v_{0}∥_{L_{ℓ+1,A_{0}}^{2}(U′′)} \\ ≤ Q_{ℓ} (∥v_{α} − v_{0}∥_{L_{2,A_{0}}^{2}(U′)} + ∥A_{α} − A_{0}∥_{L_{ℓ,A_{0}}^{2}(U′)} \\ +∥ψ_{n}(Π_{A_{α},μ}u_{α} − Π_{A_{0},μ}u_{0})∥_{L^{2}(X)}, ∥v_{α} − v_{0}∥_{L^{2}(U′)}) . \end{matrix} (8.48)

Lemma 8.17 — which we use to get convergence-to-zero for the terms ∥v_{α} − v_{0}∥_{L_{2,A_{0}}^{2}(U′)} — and the limit result (8.38) (given our choice of cutoff functions ψ_{n}) imply that the righthand side of (8.48) converges to zero as α, n → ∞. Hence, the sequence v_{α} converges to v_{0} in L_{ℓ+1,A_{0}}^{2}(U′′). The definition of the a_{α} and a_{0} implies that the sequence a_{α} converges to a_{0} in L_{ℓ,A_{0}}^{2}(U′′). Consequently, the sequence of connections A_{α} + a_{α} converges to A_{0} + a_{0} in L_{ℓ,A_{0}}^{2}(U′′). We conclude that A_{α} + a_{α} converges to A_{0} + a_{0} in L_{k,A_{0},loc}^{2}(X \ x), as desired.

The second (and final) step is to show that the sequence of measures |F (A_{α} + a_{α})|^{2} converges to |F (A_{0} + a_{0})|^{2} + 8π^{2} ∑_{x∈x} δ_{x} on X. Remark that, by hypothesis, the sequence of measures |F (A_{α})|^{2} converges to |F (A_{0})|^{2} + 8π^{2} ∑_{x∈x} δ_{x} on X. Over X \ x, the sequence of functions |F (A_{α} + a_{α})|^{2} converges in C^{∞} to |F (A_{0} + a_{0})|^{2} so, for any f ∈ C^{0}(X) and r > 0,

lim_{α→∞} \frac{1}{8π^{2}} ∫_{X\∪_{i}B(x_{i},r)} f|F (A_{α} + a_{α})|^{2} dV = \frac{1}{8π^{2}} ∫_{X\∪_{i}B(x_{i},r)} f|F (A_{0} + a_{0})|^{2} dV. (8.49)

It remains to consider the corresponding limits over the balls B(x_{i}, r).

We shall adapt the argument of [8, p. 165] and consider the Chern-Simons invariants of the restrictions of the extended anti-self-dual connections A_{0} + a_{0}, and A_{α} + a_{α} to threespheres S(x_{i}, r) = ∂B(x_{i}, r), where r is a small positive constant. Recall that if B is a connection on the product U(2) bundle over a three-manifold Y, then its Chern-Simons invariant is given by [8, p. 164]

CS_{Y} (B) := \frac{1}{8π^{2}} ∫_{Y} tr (dB + \frac{2}{3}B ∧ B ∧ B) .

If A is any extension of B to a connection over a four-manifold Z with boundary Y, then

CS_{Y} (B) = −\frac{1}{8π^{2}} ∫_{Z} tr(F_{A} ∧ F_{A}) (mod ℤ). (8.50)

Because (|F_{A}^{+}|^{2} − |F_{A}^{+}|^{2}) dV = − tr(F_{A} ∧ F_{A}) for any connection A, we see that

lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} (|F^{−}(A_{α} + a_{α})|^{2} − |F^{+}(A_{α} + a_{α})|^{2}) dV

= lim_{α→∞} CS_{S(x_{i},r)}(A_{α} + a_{α}) (mod ℤ)

= CS_{S(x_{i},r)}(A_{0} + a_{0})

= \frac{1}{8π^{2}} ∫_{B(x_{i},r)} (|F^{−}(A_{0} + a_{0})|^{2} − |F^{+}(A_{0} + a_{0})|^{2}) dV (mod ℤ).

(8.51)

By construction (7.4) of the solutions a_{α} = d_{A_{α}}^{+,∗}v_{α}, we have F^{+}(A_{α} + a_{α}) = Π_{A_{α},μ}F^{+}(A_{α} + a_{α}) and so Lemma 6.7 implies that

∥F^{+}(A_{α} + a_{α})∥_{C^{0}(X)} ≤ C∥F^{+}(A_{α} + a_{α})∥_{L^{2}(X)},

where C is a constant independent of α by Lemma 6.7 and hypothesis (8.6). Therefore, the Lebesgue bounded convergence theorem yields

lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F^{+}(A_{α} + a_{α})|^{2} dV = \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F^{+}(A_{0} + a_{0})|^{2} dV,

and so the preceding limit identity, the fact that |F_{A}|^{2} = |F_{A}^{−}|^{2} + |F_{A}^{+}|^{2} for any connection A, and (8.51) imply that

lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F (A_{α} + a_{α})|^{2} dV

lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} ((|F^{−}(A_{α} + a_{α})|^{2} − |F^{+}(A_{α} + a_{α})|^{2}) + 2|F^{+}(A_{α} + a_{α})|^{2}) dV

= \frac{1}{8π^{2}} ∫_{B(x_{i},r)} ((|F^{−}(A_{0} + a_{0})|^{2} − |F^{+}(A_{0} + a_{0})|^{2}) + 2|F^{+}(A_{0} + a_{0})|^{2}) dV

= \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F (A_{0} + a_{0})|^{2} dV (mod ℤ).

(8.52)

Note that F (A_{α} + a_{α}) = F (A_{α}) + d_{A_{α}}a_{α} + a_{α} ∧ a_{α}, so we have the estimate

∫_{X} ||F (A_{α} + a_{α})|^{2} − |F (A_{α})|^{2}| dV
≤ 2 ∫_{X}(F (A_{α}), d_{A_{α}}a_{α} + a_{α} ∧ a_{α}) dV + 2 ∫_{X} |d_{A_{α}}a_{α} + a_{α} ∧ a_{α}|^{2} dV.

Therefore,

∫_{X} ||F (A_{α} + a_{α})|^{2} − |F (A_{α})|^{2}| dV
≤ 2(∥F (A_{α})∥_{L^{2}(X)} + ∥a_{α}∥_{L_{1,Aα}^{2}(X)})∥a_{α}∥_{L_{1,Aα}^{2}(X)}.

The terms ∥F (A_{α})∥_{L^{2}(X)}^{2} are bounded by a universal constant times ∥F (A_{0})∥_{L^{2}(X)}^{2} + 8π^{2}ℓ. The terms ∥a_{α}∥_{L_{1,Aα}^{2}(X)} are bounded by a universal constant times ∥F^{+}(A_{α})∥_{L^{♯,2}(X)}, by our a priori estimates of Lemma 7.4 for our solutions to the anti-self-dual equation (8.1). Similar remarks apply to the connections A_{0} and A_{0} + a_{0}, so we obtain

\begin{aligned}\frac{1}{8π^{2}} ∫_{X} ||F (A_{α} + a_{α})|^{2} − |F (A_{α})|^{2}| dV & ≤ \frac{1}{4}, \\ \frac{1}{8π^{2}} ∫_{X} ||F (A_{0} + a_{0})|^{2} − |F (A_{0})|^{2}| dV & ≤ \frac{1}{4}, \end{aligned} (8.53)

as our hypotheses (8.6) imply that the factors sup_{α} ∥F^{+}(A_{α})∥_{L^{2}(X)} and ∥F^{+}(A_{0})∥_{L^{2}(X)} are small. Since

lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F (A_{α})|^{2} dV = \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F (A_{0})|^{2} dV + κ_{i}, (8.54)

it follows from the limit (8.52), the estimates (8.53), and the limit (8.54) that

\begin{matrix}lim_{α→∞} \frac{1}{8π^{2}} ∫_{B(x_{i},r)} |F (A_{α} + a_{α})|^{2} dV \\ = \frac{1}{8π^{2}} ∫_{B(x_{i},r)} (|F (A_{0} + a_{0})|^{2} dV + 8π^{2}κ_{i}δ_{x_{i}}) . \end{matrix} (8.55)

Finally, observe that for all α and f ∈ C^{0}(X) we have

\frac{1}{8π^{2}} ∫_{B(x_{i},r)} |f(x_{i}) − f(x)||F (A_{α} + a_{α})|^{2} dV (x) ≤ κ sup_{x∈B(x_{i},r)} |f(x_{i}) − f(x)|, (8.56)

where κ = −\frac{1}{4}p_{1}(𝔤_{E}). By choosing a small enough constant r > 0, we can make the righthand side of (8.56) arbitrarily small, uniformly with respect to α. Hence, combining the limit identities (8.49) and (8.55) with the estimate (8.56) yields

lim_{α→∞} \frac{1}{8π^{2}} ∫_{X} f|F (A_{α} + a_{α})|^{2} dV = \frac{1}{8π^{2}} ∫_{X} f|F (A_{0} + a_{0})|^{2} dV + ∑_{i=1}^{m} κ_{i}f(x_{i}),

as required, where x ∈ Sym^{ℓ}(X) is represented by the m-tuple (x_{1}, …, x_{m}). The existence of the limit asserted above follows from its existence mod ℤ via the application of the Chern-Simons identity (8.50) in (8.51) and the curvature estimates (8.53).

9. Diffeomorphisms of moduli spaces defined by paths of metrics

Our goal in this section is to prove Theorem 1.1. defined by nearby metrics need not be diffeomorphic, no matter how close the metrics, it is worth pinpointing the precise nature of the difficulty. Thus, suppose f : M → [t_{1}, t_{2}] is a C^{r+1} map of compact manifolds with boundary, where f has no critical points and f(∂M) = {t_{1}, t_{2}}. Then there is a C^{r} diffeomorphism F : f^{−1}(t_{1})×[t_{1}, t_{2}] ≅ M such that f ∘F = π_{2}, where π_{2} : f^{−1}(t_{1})×[t_{1}, t_{2}] → [t_{1}, t_{2}] is the projection [31, Theorem 6.2.2]. The problem is that we cannot directly apply this standard result to a projection π : M_{κ}^{w,∗}(X, γ) → [t_{1}, t_{2}], because M_{κ}^{w,∗}(X, γ) is not compact. Now Theorem 6.2.2 in [31] is true for non-compact manifolds M under the extra hypothesis that M has a complete Riemannian metric for which | grad f| is bounded; without this assumption there are simple counterexamples [31, Exercise 6.2.10]. In our situation, the L^{2} metric might appear to be the obvious choice, but this is certainly not complete [29]. On the other hand we suspect that the L_{1}^{2} metric is complete and, though we shall not prove this, our proof of Theorem 1.1 does indeed make essential use of L_{1}^{2} distance functions at a crucial point.

9.1. Vector fields and diffeomorphisms. As the proof of Theorem 1.1 is rather involved, it is helpful to begin with the relatively simple underlying idea. Let Ξ be a C^{∞} vector field on a manifold 𝔐. Let D_{Ψ} ⊂ 𝔐 × ℝ be the set of points (m, t) ∈ 𝔐 × ℝ such that there exists an integral curve Ψ(m, ·) : I_{m} → 𝔐 of Ξ, where I_{m} ⊂ ℝ is an open interval with 0 ∈ I_{m} and Ψ(m, 0) = m. Recall that Ξ is complete if D_{Ψ} = 𝔐 × ℝ; if 𝔐 is compact, then Ξ is necessarily complete. We shall make use of the following basic existence result for the flow of a vector field on a Banach manifold:

Theorem 9.1. [1, Theorem 4.1.13 & Proposition 4.1.17] Let Ξ be a C^{∞} vector field on a manifold 𝔐 and let D_{Ψ} ⊆ 𝔐 × ℝ be as above. Then

1. {0} × 𝔐 ⊂ D_{Ψ}.

2. D_{Ψ} is an open subset of 𝔐 × ℝ.

3. There is a unique C^{∞} map Ψ : D_{Ψ} → 𝔐, (m, t) ↦ Ψ(m, t), such that Ψ(m, ·) is an integral curve for Ξ through m, for all m ∈ 𝔐.

4. Suppose there are constants T^{−} < 0 < T^{+} and an open interval I = (T^{+}, T^{−}) such that 𝔐 × I ⊂ D_{Ψ}. If a map Ψ_{t} : 𝔐 → 𝔐 is defined by m ↦ Ψ(m, t), t ∈ I, then Ψ_{t}(𝔐) is an open subset of 𝔐 and Ψ_{t} is a diffeomorphism from 𝔐 onto its image.

Proof. The first three assertions simply restate the corresponding assertions of Proposition 4.1.17 in [1]. The last assertion follows from Theorem 4.1.13 and Definition 4.1.10 (iv) in [1].

Let P be a Banach manifold, let π : 𝔐 → P be a C^{∞} map, and let γ : I → P be a C^{∞} curve with γ(0) = p_{0} and I ⊆ ℝ an open interval. Let γ^{−1} : C = γ(I) → I be the diffeomorphism γ(t) ↦ t. Assume that every p ∈ C ⊂ P is a regular value of π, so that π^{−1}(C) ⊂ 𝔐 is a C^{∞} submanifold. We thus obtain a function

γ^{−1} ∘ π : π^{−1}(C) ⊂ 𝔐 → ℝ, m ↦ γ^{−1}(π(m)).

Require that 𝔐 have a Riemannian metric ⟨ , ⟩, so we can define a gradient vector field Ξ := grad(γ^{−1} ∘ π) by

⟨ · , grad(γ^{−1} ∘ π)⟩ := d(γ^{−1} ∘ π).

Let Ψ : D_{Ψ} ⊆ π^{−1}(C) × I → 𝔐 be the corresponding gradient flow and assume

D_{Ψ} ⊃ π^{−1}(p_{0}) × I. (9.1)

Then, provided condition (9.1) holds, a standard argument (see the proof of Theorem 6.2.2 in [31]) implies that the map

Ψ : π^{−1}(p_{0}) × I → π^{−1}(C), (m, t) ↦ Ψ(m, t), (9.2)

is a diffeomorphism such that

(γ^{−1} ∘ π) ∘ Ψ = π_{2}, (9.3)

where π_{2} : π^{−1}(p_{0}) × I → I is projection onto the second factor. The commutativity condition (9.3) implies that the map (9.2) restricts to a diffeomorphism

(9.4) Ψ_{t} = Ψ(·, t) : π^{−1}(p_{0}) ≅ π^{−1}(p_{t}), for t ∈ I, where p_{t} = γ(t),

of the fibers of the map π : π^{−1}(C) ⊂ 𝔐 → C ⊂ P. In summary:

Proposition 9.2. Let 𝔐, P be C^{∞} manifolds, let 𝔐 have a Riemannian metric, let π : 𝔐 → P be a C^{∞} map, and let γ : I → P be a C^{∞} curve, where I ⊆ ℝ is an open interval about the origin, γ(I) = C ⊂ P, and γ(0) = p_{0}. If

• The map π is transverse to C, and

• The domain D_{Ψ} ⊆ π^{−1}(C) × I of the flow Ψ of the gradient vector field Ξ of γ^{−1} ∘

π|_{π^{−1}(C)} : π^{−1}(C) → ℝ contains π^{−1}(p_{0}) × I, then Ψ yields the diffeomorphisms (9.2) and (9.4).

9.2. Existence of the diffeomorphism, I: transversality and definition of the gradient vector field. We shall prove the first part of Theorem 1.1, which asserts the existence of the diffeomorphism M_{κ}(S^{4}, g_{0}) ≅ M_{κ}(S^{4}, g) for a metric g which is C^{1}-close enough to the standard metric g_{0}, by showing that we can satisfy the hypotheses of Proposition 9.2 when X = S^{4}, P ⊂ Met(X) is a small enough C^{1}-open neighborhood of g_{0}, 𝔐 = M_{κ}(X, C) with the L^{2} metric, π is the restriction of the projection B_{κ}(X) × P → P, and γ(t) = (1−t)g_{0}+tg, t ∈ ℝ. In this subsection we check the first condition of Proposition 9.2 (concerning transversality), while we verify the second one (concerning flow duration) in §9.3. In §9.4 and §9.5 we verify that the diffeomorphism of the fibers extends to a smoothly stratified diffeomorphism M̄_{κ}(S^{4}, g_{0}) ≅ M̄_{κ}(S^{4}, g) of their Uhlenbeck compactifications.

9.2.1. Identification of the gradient vector field. Let us initially relax the constraint that (X, g) ≅ (S^{4}, g_{0}) and allow (X, g) to be any closed, oriented, Riemannian, smooth fourmanifold, let Met(X) be the Banach space of C^{r} Riemannian metrics on X, with r ≥ 1, let γ : I_{g} → Met(X) be a C^{∞} path through g. By hypothesis, Ξ is the C^{∞} gradient vector field grad(γ^{−1} ∘ π): as we shall see below, Ξ can be taken to be the restriction of a C^{∞} vector field defined on an open subset of B_{κ}^{w,∗}(X) × Met(X). For each ([A], g) in the domain of Ξ, there is a unique maximal open interval I_{([A],g)} ⊂ ℝ about the origin and a C^{∞} flow line Ψ([A], g, ·) : I_{([A],g)} → B_{κ}^{w,∗}(X) × Met(X) such that

\frac{dΨ}{dt} = Ξ(Ψ([A], g, t)), t ∈ I_{([A],g)}, and Ψ([A], g, 0) = ([A], g). (9.5)

We let D_{Ψ} ⊂ B_{κ}^{w,∗}(X)×Met(X)×ℝ be the open subset {([A], g, t) ∈ B_{κ}^{w,∗}(X)×Met(X)×ℝ : t ∈ I_{([A],g)}}. The flow generated by Ξ is the C^{∞} map Ψ : D_{Ξ} → B_{κ}^{w,∗}(X) × Met(X). The projection A_{κ}^{w,∗}(X, g) → B_{κ}^{w,∗}(X, g) is a principal G_{E} bundle, with a canonical connection defined by the L^{2} metric induced by the Riemannian metric g on X [8, §5.4] [29], [30], so the vector field Ξ has a unique horizontal lift Ξ̃ on A_{κ}^{w,∗}(X) × Met(X) [34, Proposition 2.1.2]. The integral curve Ψ̃(A, g, ·) in A_{κ}^{w,∗}(X)×Met(X) through (A, g) is the unique lift of the integral curve Ψ([A], g, ·) in B_{κ}^{w,∗}(X) × Met(X) through ([A], g) [34, Proposition 2.3.1]. When no confusion can arise, we shall not distinguish between vector fields and flows on the base and their lifts to the total space.

Let P_{+}(g) = \frac{1}{2}(1 + ∗_{g}) : Λ^{2}(T^{∗}X) → Λ^{+,g}(T^{∗}X) be orthogonal projection. The parameterized moduli space 𝔐_{κ}^{w,∗}(X) := {([A], g) ∈ B_{κ}^{w,∗}(X) × Met(X) : P_{+}(g)F_{A} = 0} is a C^{∞} manifold by [8, Theorem 4.3.17] with tangent space at a point ([A], g) represented by

{(b, h) ∈ Ker d_{A}^{∗,g} ⊕ T_{g} Met(X) : P_{+}(g)d_{A}b + (DP_{+})_{g}(h)F_{A} = 0}. (9.6)

We can now identify the gradient vector field Ξ on M_{κ}^{w,∗}(X, C) := π^{−1}(C) ⊂ 𝔐_{κ}^{w,∗}(X) induced by the path γ : I_{g} → C ⊂ Met(X).

Lemma 9.3. Continue the above notation. Then for all ([A], g) ∈ M_{κ}^{w,∗}(X),

Ξ_{(A,g)} = (−d_{A}^{+,∗,g}(d_{A}^{+,g}d_{A}^{+,∗,g})^{−1}(DP_{+})_{g}(h)F_{A}, h) , h = ġ ∈ T_{g}C.

Proof. By definition of π, the differential

D_{(A,g)}(γ^{−1} ∘ π) : T_{(A,g)}M_{κ}^{w,∗}(X, C) → ℝ,

is just the composition of

• The restriction of the linear projection T_{A}B_{κ}^{w,∗}(X) ⊕ T_{g}P → T_{g}P, (b, h) ↦ h, to a projection T_{(A,g)}M_{κ}^{w,∗}(X, C) → T_{g}C, where T_{A}B_{κ}^{w,∗}(X) = Ker d_{A}^{∗,g} ⊂ L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}), and

• The differential (Dγ^{−1})_{g} : T_{g}C ≅ ℝ. By definition of Ξ := grad(γ^{−1} ∘ π) we have, for all (b, h) ∈ T_{(A,g)}M_{κ}^{w,∗}(X, C),

D_{(A,g)}(γ^{−1} ∘ π)(b, h) = ⟨(b, h), (Ξ_{A,g}^{1}, Ξ_{A,g}^{2})⟩,

where we view T_{(A,g)}M_{κ}^{w,∗}(X, C) ⊂ Ker d_{A}^{∗,g} ⊕ T_{g}P. Thus, since (Dπ)_{(A,g)} is just projection onto the second factor,

(9.7) ⟨b, Ξ_{A,g}^{1}⟩ + ⟨h, Ξ_{A,g}^{2}⟩ = ⟨h, Ξ_{A,g}^{2}⟩, for all (b, h) ∈ T_{(A,g)}M_{κ}^{w,∗}(X, C).

If we take h = 0 in equation (9.6), we must have b ∈ Ker d_{A}^{+,g} and so the identity (9.7) implies that Ξ_{(A,g)}^{1} ∈ (Ker d_{A}^{+,g})^{⊥}. Since we also have Ξ_{(A,g)}^{1} ∈ Ker d_{A}^{∗,g}, we see that Ξ_{(A,g)}^{1} is uniquely determined by the condition (9.6).

9.2.2. Extension of the gradient vector field. Though the gradient vector field Ξ is initially defined on M_{κ}^{w,∗}(X, C), we shall need to consider its extension to an open neighborhood B_{κ}^{w,∗}(X, C, μ) of M_{κ}^{w,∗}(X, C) in B_{κ}^{w,∗}(X) × C. We use this extension in §9.5 when we consider the vector field Ξ and its induced flow Ψ for approximately anti-self-dual connections produced by cutting off: these considerations are required for the proof of the Uhlenbeck continuity assertion of Theorem 1.1.

For a Riemannian connection A on 𝔤_{E} and Riemannian metric g on X, we recall from §6 that ν_{2}[A, g] is the least eigenvalue of the Laplacian d_{A}^{+,g}d_{A}^{+,∗} on L^{2}(Λ^{+} ⊗ 𝔤_{E}). Given a small positive constant μ, we define

B_{κ}^{w,∗}(X, C, μ) := {[A, g] ∈ B_{κ}^{w,∗}(X) × C : ν_{2}[A, g] > μ}, (9.8)

Given our C^{∞} curve γ : I_{g} ⊂ ℝ → C ⊂ Met(X) through g at t = 0, we can define an induced C^{∞} vector field Ξ on B_{κ}^{w,∗}(X, C, μ) by setting Ξ_{(A,g)} = (b, h), where h := ġ ∈ S^{2}(T^{∗}X), the Banach space of C^{r}, symmetric, rank-two, contravariant tensor fields on X, and

\begin{aligned}b(A, g) & := d_{A}^{+,∗,g}w(A, g) ∈ L_{k}^{2}(Λ^{1} ⊗ 𝔤_{E}). \\ w(A, g) & := −(d_{A}^{+,g}d_{A}^{+,∗,g})^{−1}(DP_{+})_{g}(h)F_{A} ∈ L_{k+1}^{2}(Λ^{+,g} ⊗ 𝔤_{E}). \end{aligned} (9.9)

Note that d_{A}^{∗,g}b(A, g) = 0 when P_{+}(g)F_{A} = 0. The vector field Ξ can be regarded as a lift of the vector field ġ on C ⊂ Met(X). It has a C^{∞} flow Ψ with domain an open subset D_{Ξ} ⊂ B_{κ}^{w,∗}(X, C, μ) × ℝ.

Provided we can choose a constant μ and curve C such that

M_{κ}^{w,∗}(X, C) ⊂ B_{κ}^{w,∗}(X, C, μ), (9.10)

then the vector field Ξ on B_{κ}^{w,∗}(X, C, μ) restricts to the gradient field grad(γ^{−1} ∘ π) on M_{κ}^{w,∗}(X, C). When no confusion can arise, we shall denote this extension of the gradient field by Ξ. Plainly, the extended vector field Ξ is tangent to M_{κ}^{w,∗}(X, C) and thus M_{κ}^{w,∗}(X, C) is an invariant submanifold of Ξ.

9.2.3. Eigenvalue variation and the vector field domain. We need to show that the curve C and constant μ can be chosen so that the condition (9.10) holds: without this condition we would not be able to obtain the necessary bounds on the vector field Ξ.

By definition of ν_{2}[A, g], there is a map

ν_{2} : 𝔐_{κ}^{w}(X) → ℝ, [A, g] ↦ ν_{2}[A, g], (9.11)

which extends continuously to the completion 𝔐̄_{κ}^{w}(X) — the closure of 𝔐_{κ}^{w}(X) in ∪_{ℓ=0}^{κ}(𝔐_{ℓ}^{w}(X)× Sym^{ℓ}(X)) — by Theorem 6.1. The same holds for the map ν_{0} : 𝔐_{κ}^{w}(X) → ℝ, assigning [A, g] to the least eigenvalue ν_{0}[A, g] of the Laplacian d_{A}^{∗,g}d_{A} [14].

We call g a regular value of the projection π_{ℓ} : 𝔐_{ℓ}^{w}(X) → Met(X) if Ker d_{A}^{+,g}d_{A}^{+,∗} = 0 for all [A, g] ∈ 𝔐_{ℓ}^{w}(X), that is if [A] is a smooth point for all [A] ∈ M_{ℓ}^{w}(X, g). We call g a regular value of the projection π : 𝔐̄_{κ}^{w}(X) → Met(X) if it is a regular value of the restrictions of π to 𝔐_{ℓ}^{w}(X), for each ℓ ∈ {0, …, κ}.

If ν_{2}[A, g] > 0 for all [A, x] ∈ M̄_{κ}^{w}(X, g) then, because M̄_{κ}^{w}(X, g) is compact and ν_{2} is continuous, there is an open neighborhood of M̄_{κ}^{w}(X, g) in 𝔐̄_{κ}^{w}(X) on which ν_{2} is strictly positive. Therefore, we may choose

• A positive constant μ, and

• A C^{1}-open neighborhood P of a regular value g ∈ Met(X) of π : 𝔐̄_{κ}^{w}(X) → Met(X), such that

2μ ≤ inf_{[A,g]}{ν_{2}[A, g] : [A, x] ∈ M̄_{κ}^{w}(X, g) and g ∈ P}. (9.12)

Points in M_{ℓ}^{w}(X, g) representing connections which are flat, reducible, or twisted-reducible (see [38]) need not be smooth points, so we must ensure that they are not contained in M̄_{κ}^{w}(X, g), noting that they cannot be excluded by simply choosing g to be generic as the hypotheses of the Freed-Uhlenbeck generic metrics theorem do not in general apply to flat or reducible connections [8], [24], [38]. Our main application will be to the case X = S^{4}, so we must allow w = 0 ∈ H^{2}(X; ℤ): for the remainder of this section, we shall assume w = 0 and omit it from our notation. The trivial connection [Θ, x] ∈ M̄_{κ}(X, g) is a smooth point if and only if H^{+}(X; ℝ) = 0: thus, we assume X is simply-connected, so [Θ] is the only point represented by a flat connection, and assume H^{+}(X; ℝ) = 0 so that [Θ] is a smooth point. Reducibles in M_{κ}(X, g) are in one-to-one correspondence with the set {±α : α ∈ H^{2}(X; ℤ)} [8, Proposition 4.2.15] for any metric g, so we can exclude reducibles by requiring that H^{2}(X; ℤ) = 0, that is, if H^{−}(X; ℝ) = 0. Thus, we are left with a closed, oriented, smooth four-manifold X for which π_{1}(X) = 0 and H^{2}(X; ℤ) = 0, that is, a homotopy four-sphere. For simplicity, we therefore assume for the remainder of this section that X is the four-sphere, S^{4}, and that κ > 0.

9.2.4. Bochner formulas and the vector field domain. When X = S^{4} with its standard round metric g of radius one we can alternatively use a simple Bochner argument to show that we can choose a positive constant μ and C^{1}-open neighborhood P of g_{0} so that the conditions of (9.12) are satisfied.

Lemma 9.4. Let M be a positive constant. Then there is a positive constant δ such that the following holds. For any metric g on S^{4} such that ∥g − g_{0}∥_{C^{2}(S^{4})} ≤ M, where g_{0} is the standard metric, and

∫_{S^{4}}(|W_{g}^{+}|^{2} + |R_{g} − 1|^{2}) dV_{g} ≤ δ^{2},

the Laplacian d_{A}^{+,g}d_{A}^{∗,g} has first eigenvalue greater than or equal to 1/12 for all [A] ∈ M_{κ}(S^{4}, g).

Proof. From the Bochner formula (6.2) we have d_{A_{0}}^{+,g_{0}}d_{A_{0}}^{∗,g_{0}} = \frac{1}{2}∇_{A_{0}}^{∗,g}∇_{A_{0}} + \frac{1}{6}, since W_{g_{0}}^{+} = 0, R_{g_{0}} = 1, and F_{A_{0}}^{+,g_{0}} = 0, so ν_{2}[A_{0}, g_{0}] ≥ \frac{1}{6} for all [A_{0}] ∈ M_{κ}(S^{4}, g_{0}). For another metric g and connection A such that F_{A}^{+,g} = 0, we may write

\begin{aligned}(d_{A}^{+,g}d_{A}^{∗,g}v, v)_{L^{2}} & = \frac{1}{2}(∇_{A}^{∗,g}∇_{A}v, v)_{L^{2}} + \frac{1}{6}(v, v)_{L^{2}} \\ & − (\{W_{g}^{+}, v\}, v)_{L^{2}} + \frac{1}{6}((R_{g} − 1)v, v)_{L^{2}}. \end{aligned} (9.13)

Using Lemmas 6.5 and 6.6, we see that the second-last term in (9.13) is bounded by

|({W_{g}^{+}, v}, v)_{L^{2}}| ≤ ∥W_{g}^{+}∥_{L^{2}}∥v∥_{L^{4}}^{2} ≤ c∥W_{g}^{+}∥_{L^{2}}∥v∥_{L_{1,A}^{2}}^{2}
≤ c∥W_{g}^{+}∥_{L^{2}}(∥d_{A}^{∗,g}v∥_{L^{2}}^{2} + ∥v∥_{L^{2}}^{2})
= c∥W_{g}^{+}∥_{L^{2}}((d_{A}^{+,g}d_{A}^{∗,g}v, v)_{L^{2}} + ∥v∥_{L^{2}}^{2}).

In the same vein, the last term in (9.13) is bounded by

|\frac{1}{6}((R_{g} − 1)v, v)_{L^{2}}| ≤ c∥R_{g} − 1∥_{L^{2}}((d_{A}^{+,g}d_{A}^{∗,g}v, v)_{L^{2}} + ∥v∥_{L^{2}}^{2}).

By combining these estimates we see that

(d_{A}^{+,g}d_{A}^{∗,g}v, v)_{L^{2}} ≥ \frac{1}{2}(∇_{A}^{∗,g}∇_{A}v, v)_{L^{2}} + \frac{1}{6}(v, v)_{L^{2}}
− c(∥W_{g}^{+}∥_{L^{2}} + ∥R_{g} − 1∥_{L^{2}})(d_{A}^{+,g}d_{A}^{∗,g}v, v)_{L^{2}}
− c(∥W_{g}^{+}∥_{L^{2}} + ∥R_{g} − 1∥_{L^{2}})(v, v)_{L^{2}},

and so

ν_{2}[A, g] = inf_{v≠0} \frac{(d_{A}^{+,g}d_{A}^{∗,g}v, v)_{L^{2}}}{(v, v)_{L^{2}}}

≥ \frac{\frac{1}{6} − c(∥W_{g}^{+}∥_{L^{2}} + ∥R_{g} − 1∥_{L^{2}})}{1 + c(∥W_{g}^{+}∥_{L^{2}} + ∥R_{g} − 1∥_{L^{2}})}.

The conclusion follows for small enough δ.

9.2.5. Existence of the diffeomorphism. We say that a Riemannian metric on X is good if the corresponding moduli spaces M_{ℓ}(X), 0 ≤ ℓ ≤ κ, contain no reducible connections or points [A] with H_{A}^{2} ≠ 0. Plainly, the standard metric g_{0} on S^{4} is a good metric. Since X = S^{4}, the quotient space B_{κ}(X) contains no reducibles when κ > 0. For the remainder of this section, we fix a good metric g_{0} on X and require that the curve C ⊂ Met(X) be contained in a neighborhood

P := {g ∈ Met(X) : ∥g − g_{0}∥_{C^{1}(X,g_{0})} < ε and ∥g − g_{0}∥_{C^{2}(X,g_{0})} < K}, (9.14)

where ε is small enough that P satisfies the condition (9.12) for some positive constant μ.

The proof of the existence of the diffeomorphism asserted by Theorem 1.1 is straightforward given the following key result:

Proposition 9.5. Let I_{g} ⊂ ℝ be the maximal open interval around the origin for which the curve γ : I_{g} → C ⊂ Met(X) is defined. Let ([A], g) ∈ M_{κ}(X, C) and let I_{([A],g)} ⊂ ℝ be the maximal open interval around the origin for which the flow-line through ([A], g) of the vector field Ξ on B_{κ}(X, C, μ) is defined. Then I_{([A],g)} = I_{g}.

Proof of Assertion (1) of Theorem 1.1, assuming Proposition 9.5. We need to check the hypotheses of Proposition 9.2 when 𝔐 = M_{κ}(X, C), γ(0) = g_{0}, and P is given by (9.14). By our choice of P, the map π is transverse to C, while Proposition 9.5 ensures that the domain D_{Ψ} of the gradient flow Ψ contains M_{κ}(X, g_{0}) × I_{g_{0}}. Then Proposition 9.2 implies that the map

Ψ_{t} : M_{κ}(X, g_{0}) → M_{κ}(X, g_{t}), [A] ↦ Ψ_{t}([A], g_{0}) ≡ Ψ([A], g_{0}, t), (9.15)

defines a C^{∞} diffeomorphism for all t ∈ I_{g_{0}}, as desired.

9.3. Existence of the diffeomorphism, II: duration of the flow. Our task in this subsection is to prove Proposition 9.5: the key point, as always, to derive suitable estimates on the flow.

We remark that it is not possible to obtain global L_{1}^{p} estimates on the flow-lines A_{t}, for the flow (A_{t}, g_{t}) = Ψ(A, g, t) defined by the (lift of the) vector field Ξ in (9.9) when p > 2, although it is possible for the flow considered by Taubes in [53, §3], so the arguments here differ from those of [53]. Instead, we must make do with only global L_{1}^{2} estimates on the flow-lines A(t).

Given a point [A, g] ∈ B_{κ}(X, C, μ), the maximal interval I_{([A],g)} = (T_{([A],g)}^{+}, T_{([A],g)}^{−}) ⊂ ℝ on which the flow-line Ψ([A], g, t) = (A_{t}, g_{t}) is defined is limited by the requirements that

• A_{t} is irreducible, so d_{A_{t}}^{∗,g_{t}}d_{A_{t}} has trivial kernel,

• [A_{t}] is a smooth point, so d_{A_{t}}^{+,g_{t}}d_{A_{t}}^{+,∗,g_{t}} has trivial kernel,

• A_{t} does not bubble. The possibility that the Laplacians d_{A_{t}}^{∗,g_{t}}d_{A_{t}} and d_{A_{t}}^{+,g_{t}}d_{A_{t}}^{+,∗,g_{t}} may acquire non-trivial kernels is addressed by choosing X = S^{4} and the family of metrics P ⊂ Met(X) containing the curve C to be a C^{1}-small neighborhood of the standard metric g_{0}, as discussed in §9.2. The remaining constraint, that the connections must not bubble, is addressed in this subsection. We first need to establish the existence of a universal estimate for the L^{♯,2} norm of the curvature of a g-anti-self-dual connection.

Lemma 9.6. Let (X, g) be a C^{∞}, closed, oriented, Riemannian four-manifold and let κ ≥ 0 be an integer. Then there is a constant c = c(X, g, κ) such that ∥F_{A}∥_{L^{♯,2}(X,g)} ≤ c for all [A] ∈ M_{κ}^{w}(X, g).

Proof. By hypothesis and the usual Chern-Weil identity, we have ∥F_{A}∥_{L^{2}(X,g)} = \sqrt{8π^{2}κ}, for any [A] ∈ M_{κ}^{w}(X, g). The moduli space M_{κ}^{w}(X, g) has a finite covering by Uhlenbeck neighborhoods of the type given in Definition 3.1, so it suffices to consider one such neighborhood U. Following the notation of Definition 3.1, the compact subset Y := X − ∪_{i=1}^{m}B(x_{i}, δ) of X−x has a finite covering by N balls B(y_{j}, r) such that B(y_{j}, 4r) ⊂ Y′ := X−∪_{i=1}^{m}B(x_{i}, \frac{1}{2}δ), where N = N(X, g, κ, δ) and r = r(U) is chosen so ∥F_{A}∥_{L^{2}(B(y_{j},4r),g)} is small enough that the usual elliptic estimates hold [REF!!],

∥A − Γ_{j}∥_{L_{1,Γ}^{3}(B(y_{j},2r),g)} ≤ c∥F_{A}∥_{L^{2}(B(y_{j},4r),g)}, (9.16)

where c = c(g, r) and Γ_{j} is a flat connection over B(y_{j}, 3r) such that d_{Γ_{j}}^{∗}(A−Γ_{j}) = 0. Thus,

∥F_{A}∥_{L^{♯}(B(y_{j},r),g)} ≤ c∥F_{A}∥_{L^{3}(B(y_{j},2r),g)} (Sobolev embedding [12, Lemma 4.1])
≤ c∥A − Γ_{j}∥_{L_{Γ_{j} ,1}^{3}(B(y_{j},2r),g)}
≤ c∥F_{A}∥_{L^{2}(B(y_{j},4r),g)} (by estimate (9.16))

Hence, combining the preceding estimates over each ball B(y_{j}, r) covering Y, and noting that B(y_{j}, 4r) is contained in Y′, for j = 1, …, N, we see that

∥F_{A}∥_{L^{♯}(Y,g)} ≤ c∥F_{A}∥_{L^{2}(Y′,g)},

where c = c(g, U).

We now consider each of the balls B(x_{i}, 2δ) in turn and estimate ∥F_{A}∥_{L^{♯}(B(x_{i},δ),g)}. The L^{♯} norm on two-forms is invariant under rescalings of the metric, so we may assume without loss of generality that δ = 1. We can now apply the usual bubble-tree compactness argument [13, §4], [55, §5] to conclude that there is an Uhlenbeck neighborhood U_{i} of the type given in Definition 3.1 with X replaced by B(x_{i}, 2δ), noting that no bubbling occurs in the annuli Ω(x_{i}, \frac{3}{4}δ, 2δ). We repeat the preceding argument to bound the L^{♯} norm of F_{A} on a precompact subset of B(x_{i}, 2δ) away from the (at most κ_{i} and at least two if κ_{i} ≥ 2) points of curvature concentration. This process terminates after at most κ rescalings and so yields the desired L^{♯} bound on F_{A}.

We use the preceding universal bound on ∥F_{A}∥_{L^{♯,2}(X,g)} to obtain an estimate for the flow Ψ([A], g, t).

Lemma 9.7. Let (X, g) be a C^{∞}, closed, oriented, Riemannian four-manifold let κ > 0 be an integer, and let μ be the positive constant of (9.8). Then there is a positive constant C such that the following holds. Let [A] ∈ M_{κ}(X, g) and let Ψ([A], g, t) be a flow-line of Ξ on B_{κ}(X, C, μ) through ([A], g) (at t = 0). Then, for all t_{1}, t_{2} ∈ I_{([A],g)}, and writing Ψ([A], g, t) = ([A(t)], g(t)) = ([A_{t}], g_{t}), the following hold:

(1)  ∥A(t_{2}) − A(t_{1})∥_{L^{4}(X,g)} ≤ C|t_{2} − t_{1}| sup_{t∈[t_{1},t_{2}]} ∥ġ(t)∥_{L^{∞}(X,g)},
(2)  ∥A(t_{2}) − A(t_{1})∥_{L_{1,A,g}^{2}(X)} ≤ C(1 + |t_{2} − t_{1}| sup_{t∈[t_{1},t_{2}]} ∥ġ(t)∥_{L^{∞}(X,g)})

× |t_{2} − t_{1}| sup_{t∈[t_{1},t_{2}]} ∥ġ(t)∥_{L^{∞}(X,g)}.

Proof. Applying Proposition 7.3 to estimate our expression (9.9) for the connection component, b(A, g), of the vector field Ξ(A, g) yields

∥b(A, g)∥_{L_{1,A,g}^{2}(X)} ≤ C∥(DP_{+})_{g}(h)F_{A}∥_{L^{♯,2}(X,g)}, (9.17)

where C = C(κ, μ, g). We first derive an L^{4} bound:

∥A(t_{2}) − A(t_{1})∥_{L^{4}(X,g)} ≤ ∫_{t_{1}}^{t_{2}} ∥\frac{dA}{dt}∥_{L^{4}(X,g)} dt

= ∫_{t_{1}}^{t_{2}} ∥b(Ψ(A, g, t))∥_{L^{4}(X,g)} dt
≤ c ∫_{t_{1}}^{t_{2}} ∥b(Ψ(A, g, t))∥_{L_{1,A_{t},g}^{2}(X)} dt (by Lemma 6.5)
≤ C ∫_{t_{1}}^{t_{2}} ∥(DP_{+})_{g_{t}}(ġ)F_{A_{t}}∥_{L^{♯,2}(X,g)} dt (by (9.17))
≤ C ∫_{t_{1}}^{t_{2}} ∥(DP_{+})_{g_{t}}(ġ)∥_{L^{∞}(X,g)}∥F_{A_{t}}∥_{L^{♯,2}(X,g)} dt.

(9.18)

Using Lemma 9.6 to bound the factor ∥F_{A_{t}}∥_{L^{♯,2}(X,g)} in the last line of (9.18) then yields the desired L^{4} bound in Assertion (1). Next, we have

\begin{aligned}∥A(t_{2}) − A(t_{1})∥_{L_{1,A,g}^{2}(X)} & ≤ ∫_{t_{1}}^{t_{2}} ∥\frac{dA}{dt}∥_{L_{1,A,g}^{2}(X)} dt \\ & = ∫_{t_{1}}^{t_{2}} ∥b(Ψ(A, g, t))∥_{L_{1,A,g}^{2}(X)} dt. \end{aligned} (9.19)

Note that, applying Lemma 6.5 to obtain the second inequality below, yields

\begin{aligned}∥∇_{A}b(Ψ(A, g, t))∥_{L^{2}(X,g)} & ≤ ∥∇_{A_{t}}b(Ψ(A, g, t))∥_{L^{2}(X,g)} \\ & + ∥A_{t} − A∥_{L^{4}(X,g)}∥b(Ψ(A, g, t))∥_{L^{4}(X,g)} \\ & ≤ ∥∇_{A_{t}}b(Ψ(A, g, t))∥_{L^{2}(X,g)} \\ & + c∥A_{t} − A∥_{L^{4}(X,g)}∥b(Ψ(A, g, t))∥_{L_{1,A_{t},g}^{2}(X)} \\ & ≤ (1 + c∥A_{t} − A∥_{L^{4}(X,g)})∥b(Ψ(A, g, t))∥_{L_{1,A_{t},g}^{2}(X)}. \end{aligned} (9.20)

We now insert the vector-field estimate (9.20) into the preliminary flow estimate (9.19). We then continue exactly as in the proof (9.18) of Assertion (1) to obtain the desired L_{1,A}^{2} bound in Assertion (2) of the lemma statement.

We can now proceed to the proof of Proposition 9.5, which estimates the duration of the flow of Ξ through a point ([A], g) ∈ M_{κ}(X, C).

Proof of Proposition 9.5. Using (9.9), where we defined Ξ(A, g) = (b, h) = (b, ġ), the estimates of Proposition 7.3 and Lemma 9.6 give the universal bound

∥b(A, g)∥_{L_{1,A,g}^{2}} ≤ C∥(DP_{+})_{g}(ġ)F_{A}∥_{L^{♯,2}(X,g)} ≤ C sup_{t∈I_{g}} ∥ġ∥_{C^{0}(X,g)}, (9.21)

where C = C(P, M), with M = max{1, κ, μ, μ^{−1}}. Note that P_{+}(g(t))F (A(t)) = 0 for all t ∈ I_{g}, where Ψ([A], g, t) = ([A(t)], g(t)), since M_{κ}(X, C) is an invariant submanifold for Ξ.

Plainly I_{([A],g)} ⊂ I_{g}. We claim that I_{([A],g)} ⊃ I_{g}. Suppose I_{([A],g)} ≡ (T_{([A],g)}^{−}, T_{([A],g)}^{+}) and I_{g} = (T_{g}^{−}, T_{g}^{+}), so

T_{g}^{−} ≤ T_{([A],g)}^{−} < T_{([A],g)}^{+} ≤ T_{g}^{+}.

Consider the case T_{([A],g)}^{+} < T_{g}^{+}. Let {t_{β}} ⊂ I_{([A],g)} be a sequence such that t_{β} → t_{∞} ≡ T_{([A],g)}^{+}. For all t_{α} < t_{β}, Lemma 9.7 yields

∥A(t_{β}) − A(t_{α})∥_{L_{1,A,g}^{2}} ≤ C|t_{β} − t_{α}|,

where C is defined by Assertion (2) of Lemma 9.7 and in particular, is independent of α, β. Hence, the sequence {A(t_{β})} is L_{1,A,g}^{2}-Cauchy and so converges to some L_{1}^{2} connection Ã_{∞} on 𝔤_{E_{∞}} such that P_{+}(g_{∞})F (Ã_{∞}) = 0. Here, g_{∞} = lim_{n→∞} g(t_{β}) = g(t_{∞}), where the convergence is C^{r}. In particular, there is a sequence {u_{α}} ⊂ G_{E} of L_{k+1}^{2} gauge transformations of E such that the sequence of L_{k}^{2} connections, {u_{α}(A(t_{α}))} on 𝔤_{E}, converges strongly to an L_{k}^{2} connection A_{∞} on 𝔤_{E_{∞}} such that P_{+}(g_{∞})F (A_{∞}) = 0. The flow line Ψ([A_{∞}], g_{∞}, · + t_{∞}) is defined on a open interval I_{([A_{∞}],g_{∞})} about the origin. Thus, Ψ([A_{∞}], g_{∞}, ·) gives a C^{∞} extension of the flow-line Ψ([A], g, ·) from (T_{([A],g)}^{−}, T_{([A],g)}^{+}) to an open interval in ℝ which contains (T_{([A],g)}^{−}, T_{([A],g)}^{+}]. This contradicts the maximality of T_{([A],g)}^{+}, so we must have T_{([A],g)}^{+} = T_{g}^{+}. Similarly, we must have T_{([A],g)}^{−} = T_{g}^{−}. This completes the proof.

Remark 9.8. There is a possibly a simpler proof that the sequence [A(t_{α}), g(t_{α})] ∈ M_{κ}(X, C) converges to a limit [A_{∞}, g_{∞}] ∈ M_{κ}(X, C) (where A_{∞} is an L_{k}^{2} connection, anti-self-dual with respect to a C^{r} metric g_{∞}), as α → ∞. The argument is also used to prove Lemma 9.9, so we include it here. According to the extension of the usual Uhlenbeck compactness theorem given by Proposition 9.1.2 in [8], the sequence [A(t_{α}), g(t_{α})] converges to some [A_{∞}, x, g_{∞}] ∈ M̄_{κ}(g_{∞}) × {g_{∞}}, where x ∈ Sym^{ℓ}(X) is represented by an m-tuple (x_{1}, …, x_{m}) and the x_{i} have integer multiplicities κ_{i} ≥ 0. We claim that ℓ = 0, so κ_{i} = 0 for all i. Write A_{t} = A+a_{t}, so F (A_{t}) = F_{A} +d_{A}a_{t} +a_{t} ∧a_{t} = F_{A} +d_{A_{t}}a_{t}. By Lemma 9.7 we have ∥a_{t}∥_{L_{1,A_{t}}^{2}(X)} ≤ ε, for some universal constant ε ≪ 1 such that C|t − t_{0}|∥ġ∥_{C^{0}(X,g)} ≤ ε for all t ∈ I_{g} ∩ [t_{0}, T_{g}^{+}), for t_{0} close enough to T_{g}^{+}. In particular,

∥F (A_{t})∥_{L^{2}(B(x_{i},r))} = ∥F_{A} + d_{A_{t}}a_{t}∥_{L^{2}(B(x_{i},r))}
≤ ∥F_{A}∥_{L^{2}(B(x_{i},r))} + ∥a_{t}∥_{L_{1,A_{t}}^{2}(B(x_{i},r))}
≤ cr^{2}∥F_{A}∥_{L^{∞}(B(x_{i},r))} + ∥a_{t}∥_{L_{1,A_{t}}^{2}(X)},

and therefore,

∥F (A_{t})∥_{L^{2}(B(x_{i},r))} ≤ 2ε, 0 < r < r_{0}[A], t ∈ I_{g} ∩ [t_{0}, T_{g}^{+}), (9.22)

provided r_{0}[A] is chosen small enough that cr_{0}^{2}∥F_{A}∥_{L^{∞}(B(x_{i},r_{0}))} ≤ ε. On the other hand, by Uhlenbeck convergence,

lim_{α→∞} ∫_{B(x_{i},r)} |F (A(t_{α}))|^{2} dV = ∫_{B(x_{i},r)} |F (A_{∞})|^{2} dV + 8π^{2}κ_{i},

where, as in the proof of Proposition 9.5, we suppose the sequence {t_{α}} ⊂ I_{([A],g)} converges to T_{g}^{+}. Now, for r_{0}([A], [A_{∞}]) small enough that cr_{0}^{2}∥F (A_{∞})∥_{L^{∞}(X)} ≤ ε,

∥F (A_{∞})∥_{L^{2}(B(x_{i},r))} ≤ cr^{2}∥F (A_{∞})∥_{L^{∞}(X)} ≤ ε, 0 < r < r_{0}([A], [A_{∞}]).

Therefore, given 0 < r ≤ r_{0}([A], [A_{∞}]), there is a positive constant t_{r} < T_{g}^{+} such that for all t ∈ [t_{r}, T_{g}^{+}),

∥F (A_{t})∥_{L^{2}(B(x_{i},r))}^{2} ≥ 8π^{2}κ_{i} − ε^{2},

which contradicts the upper bound obtained in the inequality (9.22) unless κ_{i} = 0. Hence, the sequence [A(t_{α}), g(t_{α})] converges to a limit [A_{∞}, g_{∞}], as desired.

9.4. Uhlenbeck continuity, I: equality of the singular supports. It remains to prove Assertion (2) of Theorem 1.1, namely that our diffeomorphism extends continuously to the Uhlenbeck compactification, namely that if [A_{α}] ∈ M_{κ}(X, g) is a sequence which converges to [A_{∞}, x] ∈ M_{κ−ℓ′}(X, g) × Sym^{ℓ}(X), then [A_{α}(t)] ∈ M_{κ}(X, g(t)) converges to [A_{∞}(t), x] ∈ M_{κ−ℓ}(X, g) × Sym^{ℓ}(X). There are two parts to the proof, involving quite different arguments:

• Show that A_{α}(t) converges, modulo gauge transformations, in L_{k,loc}^{2}(X \ x) to A_{∞}(t), and hence that A_{α}(t) converges in the Uhlenbeck topology to (A_{∞}(t), x(t)) for some x(t) ∈ Sym^{ℓ}(X) and any t ∈ I_{g};

• Show that x(t) = x for any t ∈ I_{g}. The second part is by far the easiest to prove, so we begin with that in this subsection, while the proof of the first part is given in §9.5.

We first show that the singular supports are preserved by the cobordism covering a path of metrics.

Lemma 9.9. Continue the notation and hypotheses of Lemma 9.7. Suppose Ψ([A_{α}], g, t) is a sequence of C^{∞} flow-lines through ([A_{α}], g), at t = 0, with [A_{α}] ∈ M_{κ}(X, g) converging to a limit [A_{∞}, x] ∈ M_{κ−ℓ}(X, g) × Sym^{ℓ}(X). If Ψ([A_{α}], g, t) = ([A_{α}(t)], g(t)) and [A_{α}(t)] converges to [B_{t}, x_{t}] ∈ M_{κ−ℓ′}(X, g) × Sym^{ℓ′}(X), then ℓ′ = ℓ and x_{t} = x, for all t ∈ I_{g}.

Proof. Suppose x_{t}, x are represented by the multisets (x_{1t}, …, x_{nt}) and (x_{1}, …, x_{m}), respectively, of points with positive integer multiplicities. We first claim that

{x_{1t}, …, x_{nt}} ⊆ {x_{1}, …, x_{m}}. (9.23)

Indeed, if not, there is some x_{it} ∉ {x_{1}, …, x_{m}}. Suppose δ is a positive constant. Then I_{g} is covered by finitely many overlapping closed intervals [t_{0}, t_{0} + δ] of length δ and for all t_{1}, t_{2} in such an interval, [t_{0}, t_{0} + δ],

∥A_{α}(t_{2}) − A_{α}(t_{1})∥_{L_{1,Aα(t_{1}),g}^{2}(X)} ≤ Cδ,

where C is a constant independent of α, t_{1}, t_{2}, δ. In particular, given any positive constant ε, we may choose δ small enough that Cδ ≤ ε. It suffices to show that x_{t} = x for t ∈ [0, δ], the equality for all t ∈ I_{g} following by induction.

We can now apply the argument of Remark 9.8 comparing the curvature of A_{α}(t), t ∈ [0, δ], with A_{α} ≡ A_{α}(0) on B(x_{it}, r), where Ψ([A_{α}], g, t) = ([A_{α}(t)], g(t)), and r is chosen small enough that the ball B(x_{it}, r) contains no points of x. This argument implies that x_{it} has multiplicity zero, a contradiction. Hence, by relabeling the points if necessary, we have n ≤ m and x_{it} = x_{i} for i = 1, …, n. By interchanging the roles of A_{α}(t) and A_{α}, we see that m ≤ n and thus n = m.

Second, we claim that κ(x_{it}) = κ(x_{i}) ≡ κ_{i}, for all i. We can again apply the argument of Remark 9.8 to deduce that, for all t ∈ [0, δ] and all α,

∥F (A_{α})∥_{L^{2}(B(x_{i},r))} − ε ≤ ∥F (A_{α}(t))∥_{L^{2}(B(x_{i},r))} ≤ ∥F (A_{α})∥_{L^{2}(B(x_{i},r))} + ε,

and 8π^{2}κ_{i} − ε ≤ ∥F (A_{α})∥_{L^{2}(B(x_{i},r))}^{2} ≤ 8π^{2}κ_{i} + ε. It follows that κ(x_{it}) = κ(x_{i}).

9.5. Uhlenbeck continuity, II: convergence of the background connections. The proof of the first part of Uhlenbeck continuity, convergence of the background connections, is slightly involved because of the long chains of estimates required, so we begin by providing a brief map of our strategy:

• Show that the L^{2,loc}(X \x) difference between the connection components b(·, ·) of the vector fields Ξ(·, ·) at points (χ_{0}A_{α}(t), g(t)) and (A_{α}(t), g(t)) is small, where supp χ_{0} ⊂ X \ x and χ_{0} = 1 on a large enough precompact open subset of X \ x (Lemmas 9.10 and 9.12).

• Show the L^{2}-distance between the vectors Ξ(Ψ(χ_{0}A_{α}, g, t)) and Ξ(Ψ(A_{∞}, g, t)) is controlled by the L^{2}-distance between Ψ(χ_{0}A_{α}, g, t) and Ψ(A_{∞}, g, t) (Lemma 9.13): this is is the subject of §9.5.1.

• Use Gronwall’s inequality to show the L^{2}-distance between the flows Ψ(χ_{0}A_{α}, g, t) and Ψ(A_{∞}, g, t) is controlled by the L^{2}-distance between the ODE’s initial conditions χ_{0}A_{α} and A_{∞}, plus a small error term (Lemmas 9.14 and 9.15): this is is the subject of §9.5.2.

The concluding arguments are given in §9.5.4. Cut-off connections such as χ_{0}A_{α} are defined, given a choice of points in the fibers Fr(𝔤_{E})|_{x}, x ∈ x, by the recipe of §3.3.

9.5.1. Comparison of gradient vector fields. We begin by examining the continuity properties of the vector field Ξ with respect to Uhlenbeck limits.

Lemma 9.10. Let (X, g) be a C^{∞}, closed, oriented, Riemannian four-manifold and let κ be an integer. Let A, B be L_{4}^{2} g-anti-self-dual connections on SO(3) bundles 𝔤_{E}, 𝔤_{E′}, respectively, over X, where p_{1}(𝔤_{E}), p_{1}(𝔤_{E′}) ≥ −4κ and ν_{2}[A, g], ν_{2}[B, g] ≥ 2μ. Suppose U ⊂ X is an open subset such that E|_{U} = E′|_{U}. Let ψ_{0}, χ_{0} be cutoff functions on X such that supp χ_{0} ⊂ U, supp ψ_{0} ⊂ U, and χ_{0} = 1 on supp ψ_{0}. Then

∥ψ_{0}(b(B, g) − b(A, g))∥_{L^{2}(X,g)} ≤ C∥ψ_{0}(DP_{+})_{g}(h)(F_{B} − F_{A})∥_{L^{4/3}(X,g)}
+ p(∥∇ψ_{0}∥_{L_{1}^{♯,2}(X,g)}, ∥χ_{0}(B − A)∥_{L_{1,A,g}^{2}(X)}).

where p(x, y) is a polynomial, such that p(0, 0) = 0, whose coefficients and the constant C depend only (g, ∥h(g)∥_{C^{0}(X,g)}, μ^{−1}, κ).

Proof. The result is almost immediate from the expression (9.9) for b(A, g), using Lemmas 8.5 and 8.7. We then use Lemmas 7.1 and 9.6 to give bounds for the norms

∥w(A, g∥_{L^{∞}(X,g)} + ∥w(A, g∥_{L_{2,A}^{2}(X,g)} ≤ C∥(DP_{+})(h)F_{A}∥_{L^{♯,2}(X,g)} ≤ C′,

where C = C(g, M) and M = max{1, ∥h(g)∥_{C^{0}}, κ, μ^{−1}}, and similarly for w(B, g); these terms appear as coefficients in the expression provided by Lemmas 8.5 and 8.7.

Remark 9.11. It is possible, just tedious, to explicitly write out the polynomial p(·, ·) in Lemma 9.10: one simply combines Lemmas 8.5 and 8.7.

We can use Lemma 9.10 to estimate the effect of cutting off connections on the vector field Ξ. Recall that cut-off connections such as χ_{0}A are defined, given a choice of points in the fibers Fr(𝔤_{E})|_{x}, x ∈ x, by the recipe of §3.3.

Lemma 9.12. Continue the notation of Lemma 9.10. Then,

(1) ∥ψ_{0}(b(A, g) − b(χ_{0}A, g))∥_{L^{2}(X,g)} ≤ p(∥∇ψ_{0}∥_{L_{1}^{♯,2}(X,g)}, ∥χ_{0}(1 − χ_{0})A∥_{L_{1,A}^{2}(X,g)}),
(2)  ∥χ_{0}b(A, g) − b(χ_{0}A, g)∥_{L^{2}(X,g)} ≤ p(∥∇ψ_{0}∥_{L_{1}^{♯,2}(X,g)}, ∥χ_{0}(1 − χ_{0})A∥_{L_{1,A}^{2}(X,g)})
+ ∥(1 − ψ_{0})(χ_{0}b(A, g) − b(χ_{0}A, g))∥_{L^{2}(X,g)}.

where p(x, y) is a polynomial, with coefficients depending only (g, μ, κ), such that p(0, 0) = 0, and χ_{0}(1 − χ_{0})A is regarded as an element of L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E}).

Proof. On supp ψ_{0} we have χ_{0} = 1 and thus, setting B = χ_{0}A in Lemma 9.10, we have χ_{0}(B − A) = χ_{0}(χ_{0} − 1)A, and F_{A} = F_{χ_{0}A} on supp ψ_{0}. Hence, Assertion (1) follows from Lemma 9.10. Next, observe that

∥χ_{0}b(A, g) − b(χ_{0}A, g)∥_{L^{2}} ≤ ∥ψ_{0}(χ_{0}b(A, g) − b(χ_{0}A, g))∥_{L^{2}}
+ ∥(1 − ψ_{0})(χ_{0}b(A, g) − b(χ_{0}A, g))∥_{L^{2}}
= ∥ψ_{0}(b(A, g) − b(χ_{0}A, g))∥_{L^{2}}
+ ∥(1 − ψ_{0})(χ_{0}b(A, g) − b(χ_{0}A, g))∥_{L^{2}}.

Combining this with Assertion (1) yields the desired bound in Assertion (2).

Lastly, we shall need the following version of the mean value theorem:

Lemma 9.13. Let (X, g) be a C^{∞}, closed, oriented, Riemannian four-manifold. Let M, μ be positive constants and let A be an L_{4}^{2} connection on 𝔤_{E}, with ν_{2}[A, g] ≥ 2μ. Then there are positive constants C = C([A], M, g) and δ = δ([A], g) with the following significance. If A′ is an L_{2}^{2} connection on 𝔤_{E}, with ∥A′ − A∥_{L_{1,A}^{♯,2}(X)} ≤ M and ∥A′ − A∥_{L^{4}(X)} ≤ δ, then

∥b(A′, g) − b(A, g)∥_{L^{2}(X,g)} ≤ C∥A′ − A∥_{L^{2}(X,g)}.

Proof. Let G_{A} denote the Green’s operator for the Laplacian d_{A}^{+}d_{A}^{∗} ≡ d_{A}^{+,g}d_{A}^{+,∗,g}, which has least eigenvalue ν_{2}[A, g] > 0. Since ∥A′ − A∥_{L^{4}} ≤ δ, Lemma 6.15 implies that we may choose δ > 0 small enough that d_{A′}^{+}d_{A′}^{∗} has least eigenvalue ν_{2}[A′, g] ≥ \frac{1}{2}ν_{2}[A, g] > 0. Writing A′ = A + a, we need to bound the difference

b(A + a, g) − b(A, g) = d_{A+a}^{+,∗}G_{A+a}(DP_{+})_{g}(h)F_{A+a} − d_{A}^{+,∗}G_{A}(DP_{+})_{g}(h)F_{A},

that is,

\begin{aligned}b(A + a, g) − b(A, g) & = − ∗ (a ∧ G_{A+a}(DP_{+})_{g}(h)F_{A+a}) \\ & + d_{A}^{+,∗}(G_{A+a} − G_{A})(DP_{+})_{g}(h)F_{A+a} \\ & + d_{A}^{+,∗}G_{A+a}(DP_{+})_{g}(h)(d_{A}a + a ∧ a), \end{aligned} (9.24)

where the difference of Green’s operators is given by

\begin{aligned}G_{A+a} − G_{A} & = G_{A+a}(d_{A}^{+}d_{A}^{+,∗} − d_{A+a}^{+}d_{A+a}^{+,∗})G_{A} \\ & = G_{A}(P_{+}(g)a ∧ d_{A}^{+,∗} − d_{A}^{+} ∗ (a ∧ ·))G_{A}. \end{aligned} (9.25)

Recall that ∥a∥_{L_{1,A}^{♯,2}} := ∥a∥_{L_{1,A}^{2}} + ∥∇_{A}a∥_{L^{♯}} [12]. For any ξ ∈ Ω^{+}(𝔤_{E}) we have

∥G_{A+a}ξ∥_{L^{∞}} ≤ ∥G_{A}ξ∥_{L^{∞}} + ∥G_{A}P_{+}(g)(a ∧ d_{A}^{+,∗}G_{A}ξ)∥_{L^{∞}}
+ ∥G_{A}d_{A}^{+} ∗ (a ∧ G_{A}ξ)∥_{L^{∞}} (by identity (9.25))
≤ C∥ξ∥_{L^{♯,2}} + C∥a ∧ d_{A}^{+,∗}G_{A}ξ∥_{L^{♯,2}}
+ C∥d_{A}^{+} ∗ (a ∧ G_{A}ξ)∥_{L^{♯,2}} (by Corollary 7.2)
≤ C∥ξ∥_{L^{♯,2}} + C∥a∥_{L_{1,A}^{2}}∥d_{A}^{+,∗}G_{A}ξ∥_{L_{1,A}^{2}}
+ C(∥a∥_{L_{1,A}^{♯,2}}∥G_{A}ξ∥_{L^{∞}} + ∥a∥_{L^{2♯,4}}∥d_{A}G_{A}ξ∥_{L^{2♯,4}}).

The second inequality also uses the simple eigenvalue estimate ∥v∥_{L^{2}} ≤ ν_{2}[A]^{−1}∥d_{A}^{+}d_{A}^{+,∗}v∥_{L^{2}}. The last inequality above follows from the Hölder’s inequality and Lemma 6.5, together with the Sobolev multiplication and embedding results of [12, Lemmas 4.1 & 4.3]. Therefore,

∥G_{A+a}ξ∥_{L^{∞}} ≤ C(1 + ∥a∥_{L_{1,A}^{♯,2}})∥ξ∥_{L^{♯,2}}. (9.26)

Using the estimate (9.26), the L^{2} norm of the first term in the expression (9.24) for b(A + a, g) − b(A, g) is thus bounded by

\begin{aligned}∥a ∧ G_{A+a}(DP_{+})_{g}(h)F_{A+a}∥_{L^{2}} & ≤ ∥a∥_{L^{2}}∥G_{A+a}(DP_{+})_{g}(h)F_{A+a}∥_{L^{∞}} \\ & ≤ C∥a∥_{L^{2}}(1 + ∥a∥_{L_{1,A}^{♯,2}})∥F_{A+a}∥_{L^{♯,2}}. \end{aligned} (9.27)

The L^{2} norm of the second term in identity (9.24), courtesy of identity (9.25), is bounded by

\begin{matrix}∥d_{A}^{+,∗}(G_{A+a} − G_{A})(DP_{+})_{g}(h)F_{A+a}∥_{L^{2}} \\ = ∥d_{A}^{+,∗}G_{A}(P_{+}(g)a ∧ d_{A}^{+,∗} − d_{A}^{+} ∗ (a ∧ ·))G_{A}(DP_{+})_{g}(h)F_{A+a}∥_{L^{2}} \\ ≤ C∥a ∧ d_{A}^{+,∗}G_{A}(DP_{+})_{g}(h)F_{A+a}∥_{L^{4/3}} + C∥a ∧ G_{A}(DP_{+})_{g}(h)F_{A+a}∥_{L^{2}} \\ ≤ C∥a∥_{L^{2}}(∥d_{A}^{+,∗}G_{A}(DP_{+})_{g}(h)F_{A+a}∥_{L^{4}} + ∥G_{A}(DP_{+})_{g}(h)F_{A+a}∥_{L^{∞}}) \\ ≤ C∥a∥_{L^{2}}∥F_{A+a}∥_{L^{♯,2}}. \end{matrix} (9.28)

The first inequality above follows from the estimates ∥ξ∥_{L^{2}} ≤ C∥d_{A}^{+,∗}G_{A}ξ∥_{L^{4/3}} and ∥ξ∥_{L^{2}} ≤ C∥d_{A}^{+,∗}G_{A}d_{A}^{+}ξ∥_{L^{2}}, for any ξ ∈ Γ(Λ^{+} ⊗ 𝔤_{E}). The inequality third above follows from the estimates ∥ξ∥_{L_{1,A}^{2}} ≤ C∥d_{A}^{+,∗}G_{A}ξ∥_{L^{2}}, while Corollary 7.2 and eigenvalue estimates are again used to give bounds ∥G_{A}ξ∥_{L^{∞}} ≤ C∥ξ∥_{L^{♯,2}}. Finally, the L^{2} norm of the third term in (9.24) is bounded by

\begin{matrix}∥d_{A}^{+,∗}G_{A+a}(DP_{+})_{g}(h)(d_{A}a + a ∧ a)∥_{L^{2}} \\ ≤ ∥d_{A}^{+,∗}G_{A}(DP_{+})_{g}(h)(d_{A}a + a ∧ a)∥_{L^{2}} \\ + ∥d_{A}^{+,∗}(G_{A+a} − G_{A})(DP_{+})_{g}(h)(F_{A+a} − F_{A})∥_{L^{2}} \\ ≤ C′(∥a∥_{L^{2}} + ∥a ∧ a∥_{L^{4/3}}) + C∥a∥_{L^{2}}∥(F_{A+a} − F_{A})∥_{L^{♯,2}}, \\ ≤ C′∥a∥_{L^{2}}(1 + ∥a∥_{L^{4}}) + C∥a∥_{L^{2}}(1 + ∥a∥_{L_{1,A}^{2}})∥a∥_{L_{1,A}^{♯,2}}, \end{matrix} (9.29)

noting that ∥a ∧ a∥_{L^{4/3}} ≤ ∥a∥_{L^{4}}∥a∥_{L^{2}}. The estimate for the second term above follows in the same manner as the second term in (9.24). Combining the estimates (9.27), (9.28), and (9.29) for all terms in (9.24) (and (9.25))yields the desired conclusion.

9.5.2. Gronwall’s inequality and comparison of flows. Thus far we have compared the vector fields Ξ over precompact subsets of X \ x. We now need to integrate these estimates with respect to time, t, and convert our vector-field estimates into estimates for the connection flows over precompact subsets of X \ x. To show that the L^{2}-distance between the various flows is small when restricted such subsets, we shall use the fact that the ‘initial conditions’, A_{α} and A_{∞}, are close when restricted to precompact subsets. Hence, the key point is the dependence of the ordinary differential equations defining the flows on the initial conditions and for this, as usual, the main tool at our disposal is Gronwall’s inequality:

Lemma 9.14. [1, Lemma 4.1.7] Let f, g : [t_{1}, t_{2}] → ℝ be continuous and non-negative. Suppose there is a constant M ≥ 0 such that

f(t) ≤ M + ∫_{t_{1}}^{t_{2}} f(s)g(s) ds.

Then, for all t ∈ (t_{1}, t_{2}),

f(t) ≤ M exp (∫_{t_{1}}^{t} g(s) ds) .

We now come to the application of Gronwall’s inequality.

Lemma 9.15. Continue the notation and hypotheses of Lemma 9.10. Let M < ∞ be a constant. Given [A], g, there is a constant δ such the following holds. Require that the connections A, B be g-anti-self-dual with ν_{2}[A, g] ≥ 2μ and ν_{2}[B, g] ≥ 2μ, that ∥χ_{0}B − A∥_{L_{1,A}^{2}} ≤ δ. Let K = K(κ, μ) be the constant in Lemma 9.7 and require that the metric flow g(t), t ∈ I_{g} satisfy

K|I_{g}| sup_{t∈I_{g}} ∥ġ(t)∥_{L^{∞}(X,g)} ≤ δ, (9.30)

where |I_{g}| is the length of the interval I_{g}. Then, for all t ∈ I_{g}, the following hold:

∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L_{1,A,g}^{2}(X)}

≤ ∥χ_{0}B − A∥_{L_{1,A,g}^{2}(X)} + Ct sup_{s∈I_{g}} 𝔭(s) (1)

+ C ∫_{0}^{t} ∥χ_{0}Ψ(B, g, s) − Ψ(A, g, s)∥_{L^{2}(X,g)} ds,

∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L^{2}(X,g)} (2)

≤ (∥χ_{0}B − A∥_{L^{2}(X,g)} + C|I_{g}| sup_{s∈I_{g}} 𝔭(s)) exp(−C|t|),

where 𝔭 is defined by the identity in (9.31) and C = C([A], M, g), and cut-off connections such as χ_{0}B are defined, given a choice of points in the fibers Fr(𝔤_{E})|_{x}, x ∈ x, by the recipe of §3.3.

Proof. Noting that Ψ(B, g, t) = (B_{t}, g_{t}) and applying Assertion (2) of Lemma 9.12 yields, for all t ∈ I_{g},

\begin{aligned}∥χ_{0}b(B_{t}, g_{t}) − b(χ_{0}B_{t}, g_{t})∥_{L^{2}} & ≤ p(∥∇ψ_{0}∥_{L_{1}^{♯,2}}, ∥χ_{0}(1 − χ_{0})B_{t}∥_{L_{1,B_{t}}^{2}} ) \\ & + ∥(1 − ψ_{0})(χ_{0}b(B_{t}, g_{t}) − b(χ_{0}B_{t}, g_{t}))∥_{L^{2}} \\ & := 𝔭(χ_{0}, ψ_{0}, B, g, t), \end{aligned} (9.31)

where p(x, y) is a polynomial, with coefficients depending only (g, μ, κ), such that p(0, 0) = 0, and χ_{0}(1 − χ_{0})B_{t} is regarded as an element of L_{k}^{2}(X, Λ^{1} ⊗ 𝔤_{E}).

Recalling the form of Gronwall’s inequality in Lemma (9.14), we define

f(t) := ∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L^{2}}, t ∈ I_{g}, (9.32)

and observe that, as Ψ(A, g, 0) = (A, g) and Ψ(B, g, 0) = (B, g), the definition of the flow Ψ and the vector field Ξ, together with inequality (9.31), gives

f(t) = ∥χ_{0}B − A + ∫_{0}^{t}(χ_{0}Ξ(Ψ(B, g, s)) − Ξ(Ψ(A, g, s))) ds∥_{L^{2}}
≤ ∥χ_{0}B − A∥_{L^{2}} + ∫_{0}^{t} ∥χ_{0}b(Ψ(B, g, s)) − b(Ψ(A, g, s))∥_{L^{2}} ds
≤ ∥χ_{0}B − A∥_{L^{2}}
+ ∫_{0}^{t} (𝔭(s) + ∥b(χ_{0}Ψ(B, g, s)) − b(Ψ(A, g, s))∥_{L^{2}}) ds.

(9.33)

To apply Gronwall’s inequality, we need to bound ∥b(χ_{0}Ψ(B, g, s)) − b(Ψ(A, g, s))∥_{L^{2}} by a multiple of the term f(s) = ∥χ_{0}Ψ(B, g, s) − Ψ(A, g, s)∥_{L^{2}}. To achieve this, we shall apply Lemma 9.13 after checking that its hypotheses are satisfied in this case via the following claim:

Claim 9.16. For the metric g, gauge-equivalence class [A], and a positive constant K, there are positive constants ε, M such that the following holds. Continue the above notation. Suppose χ_{0}, B, A satisfy the bounds

∥χ_{0}∥_{L_{1}^{2}}, ∥χ_{0}(B − A)∥_{L_{1,A}^{2}} ≤ K and sup_{t∈I_{g}} ∥ġ(t)∥_{L^{∞}} < ε.

Then the following hold for all t ∈ I_{g}:

(9.34)  ∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L_{1,A}^{2}} ≤ δ,
(9.35)  ∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L_{1,A}^{♯,2}} ≤ M.

Proof. We first observe that

∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L_{1,A}^{2}}
≤ ∥χ_{0}Ψ(B, g, t) − χ_{0}B∥_{L_{1,A}^{2}} + ∥χ_{0}B − A∥_{L_{1,A}^{2}}
+ ∥A − Ψ(A, g, t)∥_{L_{1,A}^{2}},
≤ ∥χ_{0}∥_{L_{1}^{2}}∥Ψ(B, g, t) − B∥_{L_{1,B}^{2}} + ∥χ_{0}∇_{B}(Ψ(B, g, t) − B)∥_{L^{2}}
+ ∥χ_{0}(B − A)∥_{L^{4}}∥Ψ(B, g, t) − B∥_{L^{4}}
+ ∥χ_{0}B − A∥_{L_{1,A}^{2}} + ∥A − Ψ(A, g, t)∥_{L_{1,A}^{2}}.
≤ (1 + ∥χ_{0}∥_{L_{1}^{2}} + ∥χ_{0}(B − A)∥_{L^{4}}) ∥Ψ(B, g, t) − B∥_{L_{1,B}^{2}}
+ ∥χ_{0}B − A∥_{L_{1,A}^{2}} + ∥A − Ψ(A, g, t)∥_{L_{1,A}^{2}}.

(9.36)

We use Lemma 9.7 to obtain bounds

\begin{aligned}∥Ψ(B, g, t) − B∥_{L_{1,B}^{2}} & ≤ δ, \\ ∥Ψ(A, g, t) − A∥_{L_{1,A}^{2}} & ≤ δ. \end{aligned} (9.37)

Consequently, substituting (9.37) into (9.36) gives the estimate (9.34).

For the inequality (9.35), the only term that might cause trouble is

∥∇_{A}(χ_{0}Ψ(B, g, t) − Ψ(A, g, t))∥_{L^{♯}} ≤ ∥∇_{A}(χ_{0}(Ψ(B, g, t) − Ψ(A, g, t)))∥_{L^{♯}}
+ ∥∇_{A}((1 − χ_{0})Ψ(A, g, t))∥_{L^{♯}}
≤ ∥∇χ_{0}∥_{L^{2♯}} (∥Ψ(B, g, t) − Ψ(A, g, t)∥_{L^{2♯}(supp ∇χ_{0})}
+∥Ψ(A, g, t)∥_{L^{2♯}(supp ∇χ_{0})})
+ ∥(1 − χ_{0})∇_{A}Ψ(A, g, t)∥_{L^{♯}(supp(1−χ_{0}))}
+ ∥χ_{0}∇_{A}(Ψ(B, g, t) − Ψ(A, g, t))∥_{L^{♯}}.

(Via the cutting-off procedure and choice of trivializations given in §3.3, the terms (1 − χ_{0})Ψ(A, g, t), Ψ(A, g, t)|_{supp ∇χ_{0}}, and Ψ(A, g, t)|_{supp(1−χ_{0})} can be regarded as 𝔤_{E}-valued oneforms over the given open subsets.) We now observe that over supp χ_{0}, the g(t)-anti-self-dual connections Ψ(B, g, t) are L_{1,A}^{2} close to the reference connections Ψ(A, g, t), for all t ∈ I_{g}; the same is true for the difference between these connections over an open subset U ⊂ X containing supp χ_{0}. Hence, our basic regularity theory from §4 yields an L_{1,A}^{3} bound for the difference Ψ(B, g, t) − Ψ(A, g, t) over supp χ_{0}, with constant depending on U — and thus χ_{0}: see, for example, our local elliptic estimates for anti-self-dual connections given by Proposition 4.20. The embedding L^{3} ⊂ L^{♯} [12, Lemma 4.1], then completes the proof of the estimate (9.35) and, hence, completes the proof of the claim.

Because we can satisfy its hypotheses using Claim 9.16, Lemma 9.13 yields, for t ∈ I_{g},

∥b(χ_{0}Ψ(B, g, t)) − b(Ψ(A, g, t))∥_{L^{2}} ≤ C∥χ_{0}Ψ(B, g, t) − Ψ(A, g, t)∥_{L^{2}}, (9.38)

where C depends on [A], M, g. Therefore, combining inequalities (9.33) and (9.38), we see that

f(t) ≤ ∥χ_{0}B − A∥_{L^{2}} + Ct sup_{s∈[0,t]} 𝔭(s)

+ C ∫_{0}^{t} ∥χ_{0}Ψ(B, g, s) − Ψ(A, g, s)∥_{L^{2}} ds

= ∥χ_{0}B − A∥_{L^{2}} + Ct sup_{s∈[0,t]} 𝔭(s) + C ∫_{0}^{t} f(s) ds.

This yields Assertion (1). With f defined by (9.32) and g the constant,

g(t) := ∥χ_{0}B − A∥_{L_{1,A,g}^{2}(X)} + C|I_{g}| sup_{s∈I_{g}} 𝔭(s), t ∈ I_{g},

we see that Assertion (2) follows from Assertion (1) and Gronwall’s inequality, Lemma 9.14.

9.5.3. Convergence of the error terms. Our application of Lemma 9.15 will, of course, be to the case where the connection B is replaced by a sequence of connections B_{α} converging to (A, x) and the cutoff functions χ_{0}, ψ_{0} are replaced by sequences χ_{n}, ψ_{n} of cutoff functions with supports in X − x and χ_{n} = 1 on supp ψ_{n}. In this situation, we need to verify that the term ∥χ_{n}B_{α} − A∥_{L^{2}(X,g)} + Ct∥𝔭_{n,α}∥_{L^{∞}([0,t])} in Lemma 9.15 converges to zero for x-good sequences of cutoff functions.

Lemma 9.17. Continue the notation and hypotheses of Lemma 9.15 and of the preceding paragraph. Let ε be a positive constant. Then for large enough n and α(n), we have |𝔭(χ_{n}, ψ_{n}, B_{α}, g, t)| < ε for all t ∈ I_{g}.

Proof. Recall from inequality (9.31) that 𝔭(t) = 𝔭(χ_{0}, ψ_{0}, B, g, t) is defined by

\begin{aligned}𝔭(χ_{0}, ψ_{0}, B, g, t) & := p(∥∇ψ_{0}∥_{L_{1}^{2}}, ∥χ_{0}(1 − χ_{0})B_{t}∥_{L_{1,B_{t}}^{2}} ) \\ & + ∥(1 − ψ_{0})(χ_{0}b(B_{t}, g_{t}) − b(χ_{0}B_{t}, g_{t}))∥_{L^{2}}, \end{aligned} (9.39)

where p(x, y) is a polynomial with p(0, 0) = 0 and coefficients depending only (g, μ, κ). We need to show that 𝔭_{n,α}(t) := 𝔭(χ_{n}, ψ_{n}, B_{α}, g, t) converges to zero for x-good sequences of cutoff functions χ_{n}, ψ_{n} and connections B_{α} converging to (A, x).

We first estimate the term ∥χ_{0}(1 − χ_{0})B_{t}∥_{L_{1,B_{t}}^{2}} appearing in the definition (9.39) of 𝔭(t).

Using ∇_{B_{t}} = ∇_{B} + (B_{t} − B), Hölder’s inequality, and the Sobolev embedding L_{1,B}^{2} ⊂ L^{4} of Lemma 6.5, we see that

\begin{aligned}∥χ_{0}(1 − χ_{0})B_{t}∥_{L_{1,B_{t}}^{2}} & ≤ ∥χ_{0}(1 − χ_{0})B_{t}∥_{L_{1,B}^{2}} + ∥(B_{t} − B) ∧ χ_{0}(1 − χ_{0})B_{t}∥_{L^{2}} \\ & ≤ ∥χ_{0}(1 − χ_{0})∥_{L_{1}^{2}}∥B_{t}∥_{L_{1,B}^{2}(supp χ_{0}(1−χ_{0}))} \\ & + ∥χ_{0}(1 − χ_{0})∇_{B}B_{t}∥_{L^{2}} \\ & + ∥B_{t} − B∥_{L^{4}}∥χ_{0}(1 − χ_{0})B_{t}∥_{L^{4}}, \end{aligned} (9.40)

where

∥χ_{0}(1 − χ_{0})∥_{L_{1}^{2}} ≤ ∥χ_{0}(1 − χ_{0})∥_{L^{2}} + ∥∇χ_{0}∥_{L^{4}}(∥χ_{0}∥_{L^{4}} + ∥1 − χ_{0}∥_{L^{4}}). (9.41)

(By the cutting-off procedure and choice of trivializations given in §3.3, the terms χ_{0}(1 − χ_{0})B_{t} and B_{t}|_{supp χ_{0}(1−χ_{0})} can be regarded as 𝔤_{E}-valued one-forms over the given open subsets.) Lemma 6.9 implies that if we replace χ_{0}, above with an x-good sequence, then the term ∥χ_{n}(1 − χ_{n})∥_{L_{1}^{2}} converges to zero as n → ∞. On the other hand, Assertion (2) of Lemma 9.7 yields the bound

∥B_{t} − B∥_{L_{1,B}^{2}} ≤ C, t ∈ I_{g}, (9.42)

where C = C(g, ∥ġ∥_{L^{∞}(I_{g})}, κ, μ). Thus, if B_{t}, B above are replaced by B_{α,t} ≡ Ψ(B_{α}, g, t) and B_{α}, respectively, the terms ∥B_{α,t} − B_{α}∥_{L_{1,Bα}^{2}} remain uniformly bounded with respect to α. Consequently, we see from from inequalities (9.40) and (9.41) that the terms ∥χ_{n}(1 − χ_{n})B_{α,t}∥_{L_{1,B_{α,t}}^{2}} on the left of inequality (9.40) converge to zero, for large enough n, α(n).

For the second term in (9.39), observe that

\begin{matrix}∥(1 − ψ_{0})(χ_{0}b(Ψ(B, g, t)) − b(χ_{0}Ψ(B, g, t)))∥_{L^{2}} \\ ≤ ∥(1 − ψ_{0})∥_{L^{4}}(∥b(Ψ(B, g, t))∥_{L^{4}} + ∥b(χ_{0}Ψ(B, g, t))∥_{L^{4}}). \end{matrix} (9.43)

From (9.17), the embedding L_{1}^{2} ⊂ L^{4} of Lemma 6.5, and the L^{♯,2} curvature bound of Lemma 9.6, we have the estimates

\begin{aligned}∥b(Ψ(B, g, t))∥_{L^{4}} & ≤ C∥(DP_{+})_{g}(ġ)F (Ψ(B, g, t))∥_{L^{♯,2}} ≤ C, \\ ∥b(χ_{0}Ψ(B, g, t))∥_{L^{4}} & ≤ C∥(DP_{+})_{g}(ġ)F (χ_{0}Ψ(B, g, t))∥_{L^{♯,2}} ≤ C. \end{aligned} (9.44)

Here we use the fact, required by Proposition 7.3, that we have a uniform bound on the L^{∞} norm of P_{+}(g_{t})F (χ_{0}Ψ(B, g, t)): this L^{∞} bound follows from the general L^{p} estimates of Proposition 5.11 for the self-dual component of the curvature of cut-off or patched antiself-dual connections, yielding a constant C depending only on (g, κ). Hence, from (9.44) we see that if we choose x-good sequences {ψ_{n}} and {χ_{n}} and if {B_{α}} converges to (A, x), then the terms ∥b(Ψ(B_{α}, g, t))∥_{L^{4}} and ∥b(χ_{n}Ψ(B_{α}, g, t))∥_{L^{4}} remain uniformly bounded with respect to α, n, while the terms ∥(1 − ψ_{n})∥_{L^{4}} converge to zero. Therefore the terms on the left in (9.43) converge to zero for large enough n, α(n).

Combining these convergence estimates for the left-hand terms in (9.40) and (9.43), we see that sup_{t∈I_{g}} 𝔭(χ_{n}, ψ_{n}, B_{α}, g, t) converges to zero for large enough n, α(n), as desired.

9.5.4. Final arguments. The assertion of continuity in Theorem 1.1 is an immediate consequence of the following result.

Proposition 9.18. Let Ψ([A_{α}], g, ·) be a sequence of C^{∞} flow-lines through ([A_{α}], g), with [A_{α}] ∈ M_{κ}(X, g) converging to a limit [A_{∞}, x] ∈ M_{κ−ℓ}(X, g)×Sym^{ℓ}(X). Then, the sequence of flow-lines Ψ([A_{α}], g, ·) in B_{κ}(X, C, μ) converges (pointwise in t ∈ I_{g}) to the C^{∞} flow-line (Ψ([A_{∞}], g, ·), x) in B_{κ−ℓ}(X, C, μ) × Sym^{ℓ}(X).

Proof. If A_{∞} is a connection on g_{E_{∞}}, we may suppose without loss of generality that the connections A_{α} converge to A_{∞} in L_{k,loc}^{2} on X − x. For each t ∈ I_{g}, the sequence Ψ([A_{α}], g, t) ≡ ([A_{α}(t)], g(t)) ∈ M_{κ}(X, g_{t}) converges to some ([A′_{t}, x_{t}], g_{t}) ∈ M̄_{κ}(X, g_{t}). We claim that ([A′_{t}, x_{t}], g_{t}) = (Ψ([A_{∞}], g, t), x), where (Ψ([A_{∞}], g, t), x) ≡ ([A_{∞}(t), x], g_{t}), for each t ∈ I_{g}.

According to Lemma 9.9, we have x_{t} = x, so it remains to show that [A′_{t}] = [A_{∞}(t)]. The standard Uhlenbeck compactness theorem [8, Theorem 4.4.3] implies that, after passing to a subsequence, the sequence of L_{k}^{2} connections A_{α}(t) converges to A′_{t} in L_{k}^{2}(U), modulo L_{k+1}^{2} gauge transformations of E, for each precompact open subset U ⋐ X \ x. Let {ψ_{n}}, {χ_{n}} be x-good sequences of cutoff functions on X, with χ_{n} = 1 on supp ψ_{n}. Then Assertion (2) of Lemma 9.15 yields, for all t ∈ I_{g},

∥χ_{n}Ψ(A_{α}, g, t) − Ψ(A_{∞}, g, t)∥_{L^{2}}

≤ (∥χ_{n}A_{α} − A_{∞}∥_{L^{2}} + Ct sup_{t∈I_{g}} 𝔭(χ_{n}, ψ_{n}, A_{α}, g, t)) exp(−C|t|).

Clearly, the terms ∥χ_{n}A_{α} −A_{∞}∥_{L^{2}} converge to zero and Lemma 9.17 implies that the terms sup_{t∈I_{g}} 𝔭(χ_{n}, ψ_{n}, A_{α}, g, t) converge to zero for large enough n and α(n). Therefore, over each U ⋐ X \ x, the sequence A_{α}(t) converges to A_{∞}(t) in L^{2}(U), modulo L_{k+1}^{2} gauge transformations of E. Hence, [A′_{t}] = [A_{∞}(t)], as desired.

We can now conclude the proof of the second assertion of Theorem 1.1.

Proof of Assertion (2) of Theorem 1.1. This is an immediate consequence of Proposition 9.18.

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Department of Mathematics, Ohio State University, Columbus, OH 43210

Current address: School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540

   E-mail address: feehan@math.ohio-state.edu and feehan@math.ias.edu

Department of Mathematics, Florida International University, Miami, FL 33199

   E-mail address: lenesst@fiu.edu