Exact triangles in Seiberg-Witten Floer theory. Part II: geometric limits of flow lines

Matilde Marcolli, Bai-Ling Wang

Contents

1 Introduction 1

2 Energy and curvature estimates 3

3 Convergence on compact sets 12 3.1 Asymptotics of finite energy ASD connections . . . . . . . . . 13 3.2 Asymptotics of monopoles on V × R . . . . . . . . . . . . . . 22 3.3 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Holomorphic disks in χ(T^{2}) . . . . . . . . . . . . . . . . . . . 31 3.5 The convergence theorem . . . . . . . . . . . . . . . . . . . . 36

4 Gluing 42 4.1 The approximate solutions . . . . . . . . . . . . . . . . . . . . 42 4.2 Linearizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.3 The solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Perturbation 59 5.1 Estimates in the perturbed case . . . . . . . . . . . . . . . . . 60 5.2 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 The boundary operators in the exact triangle 65

1 Introduction

This work is the continuation of [4]. In the previous part we analyzed the geometric limits of solutions of the Seiberg–Witten equations on a 3-manifold Y (r) with a long cylinder T^{2} × [−r, r]. We applied the result to the case of a homology sphere Y with an embedded knot K along which Dehn surgery with framing one or zero is performed to produce a homology sphere Y_{1} or a manifold Y_{0} with the homology of S^{1} × S^{2}. We proved that there is a suitable choice of metrics on Y and Y_{1}, and of a “surgery perturbation” μ of the equations on a tubular neighborhood of the knot in Y, such that the moduli spaces of solutions on Y, Y_{1}, and Y_{0} are related by

M_{Y,μ} ≅ M_{Y_{1}} ∪ ⋃_{k} M_{Y_{0}}(s_{k}),

where the s_{k} are the possible choices of Spin^{c} structures on Y_{0}. We also analyzed the splitting of the spectral flow and the relative grading of the Floer complexes on Y, Y_{1}, and Y_{0}.

In this paper we concentrate on the analysis of the geometric limits of flow lines, that is, of solutions of the Seiberg–Witten equations on the fourmanifold Y (r) × R, as r → ∞. This is necessary for an understanding of how the boundary operators in the Floer complexes of Y, Y_{1}, and Y_{0} are related.

We first consider the unperturbed equations on Y (r) × R. We derive estimates that control the convergence on compact sets. The essential phenomenon which regulates the behavior of solutions is a non-uniformity of the convergence in the non-compact time direction t ∈ R, as we stretch the length of the cylinder r → ∞. This can be seen as an effect of the presence of small eigenvalues of the linearization of the Seiberg-Witten equations on Y (r) at the two asymptotic values corresponding to t → ±∞. In fact, as already observed in [4], the linearization at these asymptotic values is a selfadjoint first order elliptic operator with small eigenvalues decaying like 1/r, as r → ∞.

We can decompose the manifold Y as Y = V ∪_{T^{2}} ν(K), where V is the knot complement and ν(K) is a tubular neighborhood of the knot. We use the same notation V and ν(K) in the following to indicate the manifolds completed with an infinite cylindrical end of the form T^{2} × [0, ∞).

We first analyze the properties of the geometric limits obtained in the convergence of solutions (A_{r}, Ψ_{r}) on fixed compact sets independent of r ≥ r_{0}. We show that on the two sides V × R and ν(K) × R these limits on compact sets decay exponentially in radial gauge along the region T^{2} × [0, ∞) × R to an asymptotic value that is (up to gauge) a constant flat connection on T^{2}.

We then analyze the convergence in asymptotic regions of the form Y (r) × ([T_{r}, ∞) ∪ (−∞, −T_{r}]), “away from compact sets”. To this purpose, we introduce a suitable rescaling of the coordinates in such a way as to report within a finite region the behavior in the complement of an arbitrarily large compact sets. After the rescaling, we obtain convergence to paths in M_{V} and M_{ν(K)}, and to holomorphic maps in χ_{0}(T^{2}, V) and χ_{0}(T^{2}, ν(K)).

These data match the geometric limits on the flat cylindrical region T^{2} × [−r, r] × R, which also consist of a flat connection, as limit uniformly on compact set, and a holomorphic map as limit after the rescaling. Thus, we can use the data of the geometric limits described above in order to form approximate solutions on Y (r) × R, for r(T, τ) depending on the rescaling parameters, used here as gluing parameters. We prove surjectivity of the linearization of the Seiberg–Witten equations at these approximate solutions. Thus, the approximate solutions can be deformed to actual solutions on Y (r) × R. In other words, we obtain a complete analogue of the gluing theorem for solutions of the 3-dimensional Seiberg–Witten monopole equations analyzed in [4].

The rescaling technique and radial gauge limits that we consider in this paper can be thought of as an analogue of the analysis of non-abelian ASD equations and holomorphic curves of [5].

Acknowledgments: We are deeply grateful to Tom Mrowka who corrected several mistakes in our understanding of the convergence and gluing of flow lines, and referred us to the results of [5] and to their rescaling technique as a possible model for our problem. The first author benefited of several conversations with T.R. Ramadas, who suggested the use of the radial gauge limits. Part of this work was done during visits of the first author to the Tata Institute of Fundamental Research in Mumbai, and to the Max Planck Institut für Mathematik in Bonn. We thank these institutions for the kind hospitality and for support. The first author is partially supported by NSF grant DMS-9802480. The second author is supported by ARC Fellowship.

2 Energy and curvature estimates

Let Y be a compact oriented smooth three-manifold with a fixed trivialization of the tangent bundle, with a local basis {e_{i}}. This trivialization determines a Spin-structure on Y, with spinor bundle S. Twisting S with a line bundle L gives a Spin^{c}-structure s, with spinor bundle W and determinant det(W) = det(S) ⊗ L^{2}. For the purpose of this paper we shall be concerned with the case of a homology sphere Y, with the unique choice of the trivial Spin^{c}-structure, or with the case of Y_{0}, a 3-manifold with the homology of S^{1} × S^{2}, with the infinite family of Spin^{c} structures s_{k} with c_{1}(det(W_{k})) = 2k, as discussed in [4].

On the manifold Y (r), consider a solution (A_{r}, ψ_{r}) of the Seiberg–Witten equations

\begin{matrix}∗ F_{A} = σ(ψ, ψ) \\ ∂_{A}ψ = 0, \end{matrix}  (1)

given in terms of a U(1)-connection A on the line bundle L, and a section ψ ∈ Γ(Y, W). The 1-form σ(ψ, ψ) is given in local coordinates by σ(ψ, ψ) = ∑_{i}⟨e_{i}.ψ, ψ⟩e^{i}. Solutions are critical points of the Chern–Simons– Dirac functional

CSD(A, ψ) = −\frac{1}{2} ∫_{Y} (A − A_{0}) ∧ (F_{A} + F_{A_{0}}) +_{∫Y} ⟨ψ, ∂_{A}ψ⟩dvol_{Y} .

For the moment, we only consider the unperturbed functional and the unperturbed equations (1). Later in this paper it will be necessary to formulate the results in the perturbed case, with the perturbations introduced in [4].

Let L_{A_{r},ψ_{r}} be the linearization of the equations, namely the operator

L_{A_{r},ψ_{r}}(α, ϕ) = (− ∗ dα + σ(ψ_{r}, ϕ), ∂_{A_{r}}ϕ + \frac{1}{2}α.ψ_{r}) .

This, together with the infinitesimal action of the gauge group and the gauge fixing condition, defines the extended Hessian of the Chern-Simons-Dirac functional, namely the first order elliptic operator

H_{A_{r},ψ_{r}}(f, α, ϕ) = (− ∗ dα + σ(ψ_{r}, ϕ) − df,
∂_{A_{r}}ϕ + \frac{1}{2}α.ψ_{r} + fψ_{r},
−d^{∗}α − iIm⟨ψ_{r}, ϕ⟩).

In [4] we proved the following gluing theorem. We assigned a metric on Y which has positive scalar curvature on a tubular neighborhood ν(K) of the knot. We proved that, with this choice of the metric, for large r ≥ R_{0}, the solutions (A_{r}, ψ_{r}) can be written as a gluing

(A_{r}, ψ_{r}) = (A′, ψ′)#_{r}(A′′, 0).

Here (A′, ψ′) is a finite energy solution of (1) on the knot complement V endowed with an infinite cylindrical end T^{2} × [0, ∞), and (A′′, 0) is a flat connection on ν(K). The length of the cylinder r ≥ R_{0} appears as the gluing parameter. The gluing happens at the asymptotic value a_{∞} of (A′, ψ′) along the end T^{2} × [0, ∞) of the knot complement V: a_{∞} is a flat connection on T^{2}, which lives in a cyclic cover χ_{0}(T^{2}, V) of the character variety χ(T^{2}). Up to a gauge transformation, a_{∞} agrees with the restriction to T^{2} of the connection A′′.

In this paper we formulate the analogous gluing theorem for the gradient flow lines of the Chern–Simons–Dirac functional, that is, for solutions of the Seiberg–Witten equations on Y (r) × R. The gluing theorem, in this case, is more complicated because of the lack of uniformity in the convergence as r → ∞. This gives rise to more complicated geometric limits. We report here a statement from [4] which is the ultimate source of the interesting non-uniform limits of flow lines that we are going to discuss in the rest of this paper.

Proposition 2.1 The dimension N(r, o(1)) of the span of eigenvectors of the operator H_{A_{r},ψ_{r}} with eigenvalues satisfying μ(r) → 0 as r → ∞ is given by

N(r, o(1)) = dim Ker_{L^{2}}(H_{(A′,ψ′)}) + dim Ker_{L^{2}}(H_{(A′′,0)}) + dim Ker(Q_{a_{∞}}),

where a_{∞} is the asymptotic value of (A′, ψ′) as s → ∞.

For any ϵ > 0, the dimension N(r, r^{−(1+ϵ)}) of the span of eigenvectors of the operator H_{A_{r},ψ_{r}} with eigenvalues μ < r^{−(1+ϵ)} is given by

N(r, r^{−(1+ϵ)}) = dim Ker_{L^{2}}(H_{(A′,ψ′)}) + dim Ker_{L^{2}}(H_{(A′′,0)}) + dim ℓ_{1} ∩ ℓ_{2},

where ℓ_{1} ∩ ℓ_{2} is the intersection of the two Lagrangian submanifolds determined by the extended L^{2} solutions of H_{(A′,ψ′)}(α, ϕ) = 0 and H_{(A′′,0)}(α, ϕ) = 0 in the tangent space H^{1}(T^{2}, iR) = Ker(Q_{a_{∞}}) of χ_{0}(T^{2}).

Thus, under the assumption that

Ker_{L^{2}}(H_{(A′,ψ′)}) = Ker_{L^{2}}(H_{(A′′,0)}) = 0,

we have N(r, o(1)) = 2, generated by the elements of H^{1}(T^{2}, iR), and N(r, r^{−(1+ϵ)}) = 0, for all ϵ > 0.

The proof of Proposition 2.1 relies on the results of [3], cf. [4].

Let Y (r) be a closed three-manifold with a long tube [−r, r] × T^{2}, endowed with the flat product metric on the tube.

Let (A_{r}, Ψ_{r}) be a finite energy solution of the Seiberg-Witten equations on the four-manifold Y (r) × R, that is, (A_{r}, Ψ_{r}) satisfies the equations

\begin{matrix}F_{A}^{+} = τ(Ψ, Ψ) \\ D_{A}Ψ = 0, \end{matrix}  (2)

with the condition

E_{r} = ∥∂_{t}A_{r}∥_{L^{2}(Y (r)×R)}^{2} + ∥∂_{t}ψ_{r}∥_{L^{2}(Y (r)×R)}^{2} < ∞.

Here (A_{r}(t), ψ_{r}(t)) is a solution in a temporal gauge, gauge equivalent to the original (A_{r}, Ψ_{r}). The self-dual form τ(Ψ, Ψ) is given in local coordinates by τ(Ψ, Ψ) = ∑_{i,j}⟨e_{i}e_{j}Ψ, Ψ⟩e^{i} ∧ e^{j}. As proved in [11], the finite energy condition ensures the existence of asymptotic values (A_{r}(±∞), ψ_{r}(±∞)) as t → ±∞, for any fixed r ≥ r_{0}. The asymptotic values satisfy the Seiberg-Witten equations (1) on Y (r).

If the asymptotic values are irreducibles, satisfying ψ_{r}(±∞) ≠ 0, then the convergence as t → ±∞ is exponential,

∥(A_{r}(t), ψ_{r}(t)) − (A_{r}(±∞), ψ_{r}(±∞))∥_{L_{1}^{2}(Y (r)×{t})} ≤ C_{r}e^{−δ_{r}|t|},  (3)

for all t ≥ T_{r}. The rate of decay δ_{r} is determined by the absolute value of the first non-trivial eigenvalue of the linearization L_{A_{r}(±∞),ψ_{r}(±∞)} of the monopole equations (1) at (A_{r}(±∞), ψ_{r}(±∞)), hence, by Proposition 2.1, the rate of decay to the asymptotic values at ±∞ satisfies

δ_{r} = \frac{c}{r} .

We are interested in studying the boundary operator of the Floer complex, hence we are only interested in flow lines that connect irreducible critical points. In the case of equivariant Floer theory [11] we need also consider flow lines connecting to the reducible point, but in the framed configuration space the reducible solution is also a smooth point and a non-degenerate critical point, hence we have the same exponential decay of flow lines, as proved in [11].

Lemma 2.2 Let (A, Ψ) be a solution of the Seiberg-Witten equations (2) on Y (r) × R, for a fixed r > 0. Suppose that (A, Ψ) is in a temporal gauge on Y (r) × [T, ∞) and Y (r) × (−∞, −T] and assume that it decays exponentially to asymptotic values (A(±∞), ψ(±∞)) as t → ±∞, where the (A(±∞), ψ(±∞)) satisfy the Seiberg-Witten equations (1).

Then we have the following identity

−\frac{1}{2} ∫_{Y (r)}(A(−∞)−A_{0})∧(F_{A(−∞)}+F_{A_{0}})+\frac{1}{2} ∫_{Y (r)}(A(∞)−A_{0})∧(F_{A(∞)}+F_{A_{0}})

= \frac{1}{2} ∫_{Y (r)×R}F_{A} ∧ F_{A}.

The identity holds more generally when replacing Y (r) with a smaller domain Ω ⊂ Y (r).

Proof. We have

−\frac{1}{2} ∫_{Y (r)}(A(−∞)−A_{0})∧(F_{A(−∞)}+F_{A_{0}})+\frac{1}{2} ∫_{Y (r)}(A(∞)−A_{0})∧(F_{A(∞)}+F_{A_{0}})

= −\frac{1}{2} ∫_{Y (r)}(A(−∞) − A_{0}) ∧ (F_{A(−∞)} + F_{A_{0}})
+\frac{1}{2} ∫_{Y (r)}(A(∞) − A_{0}) ∧ (F_{A(∞)} + F_{A_{0}})
= \frac{1}{2} ∫_{R} dt\frac{∂}{∂t} ∫_{Y (r)}((A(t) − A_{0}) ∧ (F_{A(t)} + F_{A_{0}}))
= −\frac{1}{2} ∫_{Y (r)} ∫_{R} \frac{∂}{∂t}F_{A(t)} ∧ (A(t) − A_{0}) ∧ dt
−\frac{1}{2} ∫_{Y (r)} ∫_{R}(F_{A(t)} + F_{A_{0}}) ∧ \frac{∂}{∂t}A(t) ∧ dt
= −\frac{1}{2} ∫_{Y (r)×R}d_{Y (r)}(\frac{∂}{∂t}A(t)) ∧ (A(t) − A_{0}) ∧ dt
−\frac{1}{2} ∫_{Y (r)} ∫_{R}(F_{A(t)} + F_{A_{0}}) ∧ \frac{∂}{∂t}A(t) ∧ dt
= −\frac{1}{2} ∫_{Y (r)×R}(F_{A(t)} − F_{A_{0}}) ∧ \frac{∂}{∂t}A(t) ∧ dt
−\frac{1}{2} ∫_{Y (r)×R}(F_{A(t)} + F_{A_{0}}) ∧ \frac{∂}{∂t}A(t) ∧ dt
= −_{∫Y (r)×R}F_{A(t)} ∧ \frac{∂A(t)}{∂t} ∧ dt
= \frac{1}{2} ∫_{Y (r)×R}F_{A} ∧ F_{A}.

This completes the proof. ⋄

Lemma 2.3 Let (A_{r}, Ψ_{r}) be a solution of the Seiberg-Witten equations (2) on Y (r) × R, such that the gauge transformed element

λ_{r}(A_{r}, Ψ_{r}) = (A_{r}(t), ψ_{r}(t))

in a temporal gauge satisfies the finite energy condition

E_{r} = ∥∂_{t}A_{r}∥_{L^{2}(Y (r)×R)}^{2} + ∥∂_{t}ψ_{r}∥_{L^{2}(Y (r)×R)}^{2} =

∫_{R} ∥∇CSD(A_{r}(t), ψ_{r}(t))∥_{L^{2}(Y (r))}^{2}dt < ∞.

Then, for any interval [t_{0}, t_{1}] of length ℓ = t_{1} − t_{0}, we have estimates

∥Ψ_{r}∥_{L^{4}(Y (r)×[t_{0},t_{1}])}^{4} ≤ 8E_{r} + 2s_{0}^{2}vol(Y (r_{0}))l,
∫_{t_{0}}^{t_{1}} ∥F_{A_{r}(t)}∥_{L^{2}(Y (r)×{t})}^{2}dt ≤ E_{r} + s_{0}^{2}vol(Y (r_{0}))l,

and

∫_{t_{0}}^{t_{1}} ∥∇_{A_{r}(t)}ψ_{r}(t)∥_{L^{2}(Y (r)×{t})}^{2}dt ≤ E_{r} + s_{0}^{2}vol(Y (r_{0}))l,

for all r ≥ r_{0}.

Moreover, the energy E_{r} is uniformly bounded in r ≥ r_{0}. The constant s_{0} ≥ 0 is defined by

s_{0} = max_{Y (r_{0})}{−s(x), 0},  (4)

with s(x) the scalar curvature.

Proof. The argument follows 6.12 of [16]. Since the cylinder [−r, r] × T^{2} is endowed with the flat metric, we have

s(r) = max_{Y (r)}{−s(x), 0} = s_{0},

for r ≥ r_{0} and s_{0} as in (4). Moreover, we can estimate

\begin{matrix}E_{r} ≥ ∫_{t_{0}}^{t_{1}}(\frac{1}{2}∥F_{A_{r}(t)}∥_{L^{2}(Y (r))}^{2} + ∥∇_{A_{r}(t)}ψ_{r}(t)∥_{L^{2}(Y (r))}^{2})dt+ \\ ∫_{Y (r)×[t_{0},t_{1}]} \frac{1}{8}|Ψ_{r}|^{4}dv_{Y (r)}dt − \frac{s_{0}}{4} ∫_{Y (r_{0})×[t_{0},t_{1}]} |Ψ_{r}|^{2}dv_{Y (r_{0})}dt. \end{matrix}

At any local maximum, |Ψ_{r}|^{2} is bounded by the scalar curvature, hence a non-trivial maximum can only occur away from [−r, r] × T^{2}, i.e. on Y (r_{0}) × R. Thus, the estimate above gives

∥Ψ_{r}∥_{L^{4}(Y (r)×[t_{0},t_{1}])}^{4} ≤ 8E_{r} + 4s_{0}^{2}vol(Y (r_{0}))l.

The other two estimates follow similarly.

The uniform bound on the energy E_{r} is obtained as follows. As we discussed before, for each fixed r > 0, the finite energy condition for (A_{r}, Ψ_{r}) forces the existence of asymptotic values (A_{r}(±∞), ψ_{r}(±∞)) as t → ±∞ that satisfy the 3-dim Seiberg-Witten equations (1). Moreover, if the asymptotic values are non-degenerate critical points of the Chern-Simons-Dirac functional, then the temporal gauge representative (A_{r}(t), ψ_{r}(t)) decays sufficiently fast in t to the asymptotic values in the L_{2}^{2} topology.

Thus, the energy E_{r} can be written also as the total variation of the Chern-Simons-Dirac functional along the path (A_{r}(t), ψ_{r}(t)),

E_{r} = CSD(A_{r}(−∞), ψ_{r}(−∞)) − CSD(A_{r}(∞), ψ_{r}(∞)).

Moreover, since the elements (A_{r}(±∞), ψ_{r}(±∞)) satisfy the equations (1) on Y (r), we have ∂_{A_{r}(±∞)}ψ_{r}(±∞) = 0 and the variation of the CSD functional is simply given by

E_{r} = \frac{1}{2} ∫_{Y (r)}(A_{r}(∞) − A_{0}) ∧ (F_{A_{r}(∞)} + F_{A_{0}})

−\frac{1}{2} ∫_{Y (r)}(A_{r}(−∞) − A_{0}) ∧ (F_{A_{r}(−∞)} + F_{A_{0}})

The previous Lemma shows that this quantity can be rewritten as

E_{r} = \frac{1}{2} ∫_{Y (r)×R} F_{A_{r}} ∧ F_{A_{r}}.

This is a topological term, conformal and gauge invariant, hence it does not change when stretching the cylinder [−r, r] × T^{2}. ⋄

Lemma 2.4 Let (A_{r}(t), ψ_{r}(t)) be a finite energy solution of the Seiberg-Witten equations (2) on Y (r) × R, in temporal gauge in the R-direction. Then, for all t ≥ T_{r}, we can write the estimate (3) as

∥(A_{r}(t), ψ_{r}(t)) − (A_{r}(±∞), ψ_{r}(±∞))∥_{L^{2}(Y (r)×\{t\})} ≤ Ce^{− \frac{c|t|}{r}} ,

where the constant C is independent of r. Moreover, on any fixed compact set K ⊂ Y (r), independent of r ≥ r_{0}, we have

∥(A_{r}(t), ψ_{r}(t)) − (A_{r}(±∞), ψ_{r}(±∞))∥_{L_{1}^{2}(K′×\{t\})} ≤ C_{K}e^{− \frac{c|t|}{r}} ,

with K′ ⊂ int(K), and with C_{K} independent of r.

Proof. The L^{2} bound follows by the exponential decay of the energy functional: for t > T_{r} we have

\frac{d}{dt}E_{r}(t) ≤ E_{r}(0) exp(−\frac{ct}{r}),

where

E_{r}(t) = ∫_{t}^{∞} ∥∂_{t}A_{r}∥_{L^{2}(Y (r))}^{2} + ∥∂_{t}ψ_{r}∥_{L^{2}(Y (r))}^{2}dt,

and E_{r}(0) uniformly bounded in r ≥ r_{0}, by Lemma 2.3.

The L_{1}^{2} bound follows by retracing the argument of Lemma 6.14 of [16]: we obtain that the constant C_{r} of our (3) depends on the underlying manifold Y (r) through the constant of the Sobolev multiplication theorem, and the estimates of our Lemma 2.3, which are uniform in r ≥ r_{0}. Consider a compact set K that is embedded in Y (r) for all r ≥ r_{0}, so that the metric on K does not change with r ≥ r_{0}. If we restrict our estimate to K, and we choose the cutoff function ξ of Lemma 6.14 of [16] supported in K, we obtain a uniform C_{K} depending only on K. ⋄

Lemma 2.5 Suppose given (A_{r}, Ψ_{r}), a finite energy solution of the equations (2) on Y (r) × R, for r ≥ r_{0}. There is a pointwise bound

|Ψ_{r}(x, t)|^{2} ≤ s_{0},

where s_{0} = max_{Y (r_{0})}{−s(x), 0}, with s(x) the scalar curvature. Moreover, for any t_{0} < t_{1}, we have an estimate

∫_{Y (r)×[t_{0},t_{1}]}⟨Ψ_{r}, ∇_{A_{r}}^{∗}∇_{A_{r}}Ψ_{r}⟩dv_{Y (r)}dt ≤ \frac{s_{0}}{4} ∥Ψ_{r}∥_{L^{2}(Y (r_{0})×[t_{0},t_{1}])}^{2}.

Proof. The finite energy condition ensures the existence of asymptotic values ψ_{r}(x, ±∞) for the spinor Ψ_{r}(x, t). Thus, we can estimate

|Ψ_{r}(x, t)|^{2} ≤ max{|ψ_{r}(x, ±∞)|^{2}, −s_{r}(x, t)},

with s_{r}(x, t) the scalar curvature on Y (r) × R, and the pointwise estimate follows. By the Weitzenböck formula and the equations we have

0 = ∇_{A_{r}}^{∗}∇_{A_{r}}Ψ_{r} + \frac{s_{r}}{4} Ψ_{r} + \frac{|Ψ_{r}|^{2}}{4} Ψ_{r}.

By integrating and by using the vanishing of the scalar curvature on the cylinder T^{2} × [−r, r], we obtain the desired estimate. ⋄

In the following we use the notation

E_{r}(t_{0}, t_{1}) = ∫_{t_{0}}^{t_{1}} ∥∇CSD(A_{r}(t), ψ_{r}(t))∥_{L^{2}(Y (r))}^{2}dt.

Proposition 2.6 Assume, as before, that (A_{r}, Ψ_{r}) is a finite energy solution of the equations (2) on Y (r) × R. Suppose given any two real numbers t_{0} < t_{1}. Then, there is a uniform bound on ∥F_{A_{r}}∥_{L^{2}(Y (r)×[t_{0},t_{1}])} for r ≥ r_{0},

∥F_{A_{r}}∥_{L^{2}(Y (r)×[t_{0},t_{1}])}^{2} ≤ C_{ℓ},

where C_{ℓ} is a constant independent of r and depending only on ℓ = t_{1} − t_{0}.

Proof. According to the previous Lemma we have a uniform bound on the L^{4}-norm

∥Ψ_{r}∥_{L^{4}(Y (r)×[t_{0},t_{1}])},

hence, by the equations, we obtain a uniform bound on ∥F_{A_{r}}^{+}∥_{L^{2}(Y (r)×[t_{0},t_{1}])}^{2}. We have the identity

(2|F_{A}^{+}|^{2} − |F_{A}|^{2})dv = −F_{A} ∧ F_{A}.  (5)

Thus, we obtain an estimate

∥F_{A_{r}}∥_{L^{2}(Y (r)×[t_{0},t_{1}])}^{2} ≤ 2 ∥F_{A}^{+}∥_{L^{2}(Y (r)×[t_{0},t_{1}])}^{2} + E_{r}(t_{0}, t_{1})
+ ∫_{Y (r)}⟨ψ_{r}(t_{1}), ∂_{A_{r}(t_{1})}ψ_{r}(t_{1})⟩dv_{Y (r)}
− ∫_{Y (r)}⟨ψ_{r}(t_{0}), ∂_{A_{r}(t_{0})}ψ_{r}(t_{0})⟩dv_{Y (r)}.

In the right hand side we can estimate

E_{r}(t_{0}, t_{1}) ≤ E_{r},

which is uniformly bounded in r ≥ r_{0}, by Lemma 2.3. We need to estimate the term

| ∫_{Y (r)}⟨ψ_{r}(t_{1}), ∂_{A_{r}(t_{1})}ψ_{r}(t_{1})⟩dv_{Y (r)} − ∫_{Y (r)}⟨ψ_{r}(t_{0}), ∂_{A_{r}(t_{0})}ψ_{r}(t_{0})⟩dv_{Y (r)}|.

We estimate

| ∫_{t_{0}}^{t_{1}} \frac{d}{dt} ∫_{Y (r)}⟨ψ_{r}(t), ∂_{A_{r}(t)}ψ_{r}(t)⟩dv_{Y (r)}dt| ≤
∫_{t_{0}}^{t_{1}}(2∥∂_{A_{r}(t)}ψ_{r}(t)∥_{L^{2}(Y (r))}^{2} + 2|⟨F_{A_{r}(t)}, ∗σ(ψ_{r}(t), ψ_{r}(t))⟩|)dt ≤
∫_{t_{0}}^{t_{1}} 2∥∇_{A_{r}(t)}ψ_{r}(t)∥_{L^{2}(Y (r))}^{2}+
(∫_{t_{0}}^{t_{1}} ∥F_{A_{r}(t)}∥_{L^{2}(Y (r))}^{2}dt)^{1/2}(∫_{t_{0}}^{t_{1}} ∫_{Y (r)} ∥ψ_{r}(t)∥^{4}dv_{Y (r)}dt)^{1/2}+
\frac{s_{0}}{2} ∥Ψ_{r}∥_{L^{2}(Y (r_{0})×[t_{0},t_{1}])}^{2} + \frac{1}{2}∥Ψ_{r}∥_{L^{4}(Y (r)×[t_{0},t_{1}])}^{4}.

Now the estimates of Lemma 2.3 give the required bound on ∥F_{A_{r}}∥. ⋄

3 Convergence on compact sets

In the process of stretching the neck in Y (r) × R, the spinor tends to vanish pointwise on the long cylinder, as the following Lemma proves.

Lemma 3.1 Assume, as before, that (A_{r}, Ψ_{r}) is a finite energy solution of the equations (2) on Y (r) × R. Given any compact set of the form T^{2} × [−r_{0}, r_{0}] × [t_{0}, t_{1}], we have pointwise convergence

|Ψ_{r}(x, t)| → 0,

for all (x, t) ∈ T^{2} × [−r_{0}, r_{0}] × [t_{0}, t_{1}], as r → ∞.

Proof. The claim follows from the uniform bound

∫_{T^{2}×[−r,r]×[t_{0},t_{1}]} |Ψ_{r}(x, t)|^{4}dvdt ≤ C_{ℓ}

derived previously. ⋄

Thus, the self-dual part of the curvature is also converging to zero pointwise on the long cylinder.

Corollary 3.2 If (A_{r}, Ψ_{r}) is a finite energy solution of the equations (2) on Y (r) × R, then, on compact sets of the form T^{2} × [−r_{0}, r_{0}] × [t_{0}, t_{1}], the connection A_{r} converges to a finite energy solution A of the abelian self dual equation F_{A}^{+} = 0.

Proposition 3.3 Inside the manifold Y (r) × R, for all r ≥ r_{0}, consider fixed compact sets of the form K_{V} ⊂ V × R, K_{ν} ⊂ ν(K) × R, or K_{0} ⊂ T^{2} × [−r_{0}, r_{0}] × R. We assume that the metric on ν(K) has non-negative scalar curvature, and that it is flat on the cylinder T^{2} ×[−r, r]. Let (A_{r}, Ψ_{r}) be finite energy solutions of the equations (2) on Y (r)×R. Then there exists a subsequence (A_{r′}, Ψ_{r′}) and gauge transformations λ_{r′} on Y (r′) × R, such that the sequence λ_{r′}(A_{r′}, Ψ_{r′}) satisfies the following properties. It converges smoothly on the compact sets K_{V} to a finite energy solution (A, Ψ) of the four-dimensional Seiberg-Witten equations on V × R. On the compact sets K_{ν} it converges to a finite energy solution (A, 0) of the abelian ASD equation on ν(K) × R, and on the compact sets K_{0} it converges to a finite energy solution of the abelian ASD equation on T^{2} × R^{2}.

Proof. We can assume that (A_{r}, Ψ_{r}) is in L_{k,δ_{r}}^{2}(Y (r′) × R), with δ_{r} ∼ 1/r, by the exponential decay to the asymptotic values. On a fixed compact set K the uniform pointwise bound on the spinor implies the L^{2}-bound

∥Ψ_{r}∥_{L^{2}(K)} ≤ V ol(K)s_{0}.

The uniform bound on the curvature provides a uniform L_{1}^{2}-bound on the connection (up to a gauge transformation), as in Lemma 5.3.1 of [14]. The uniform pointwise bound on the spinor, together with the uniform bound

∥∇_{A_{r}}Ψ_{r}∥_{L^{2}(Y (r)×[t_{0},t_{1}])} ≤ \frac{1}{4}s_{0}∥Ψ_{r}∥_{L^{2}(Y (r_{0})×[t_{0},t_{1}])}

provides a uniform L^{2}-bound for dF_{A_{r}}^{+}, as in Lemma 5.3.3 of [14]. A bootstrapping argument then bounds the higher Sobolev norms. ⋄

3.1 Asymptotics of finite energy ASD connections

In this subsection we study explicitly the solutions of the abelian ASD equation on T^{2} ×R^{2} and on an asymptotic end of the form T^{2} ×[0, ∞)×R. The first case will provide the geometric limit on compact sets on the cylinder T^{2} ×[−r, r]×R, as r → ∞, and the second case will give the geometric limit on compact sets on ν(K) × R, and will serve as a model for the asymptotics of the limits on V × R, as we discuss later in this work.

If we represent the connection as

A = a(w, s, t) + f(w, s, t)ds + h(w, s, t)dt,

the abelian ASD equation can be written as the pair of equations

∂_{t}a − dh + ∗(∂_{s}a − df) = 0
∂_{t}f − ∂_{s}h + ∗F_{a} = 0.

With a change of variables z = s + it and z = e^{ρ+iθ}, we can write

a(w, ρ, θ) = a(w, e^{ρ+iθ})
f(w, ρ, θ) = e^{−ρ} cos θ h(w, e^{ρ+iθ}) − e^{−ρ} sin θ f(w, e^{ρ+iθ})
h(w, ρ, θ) = e^{−ρ} cos θ f(w, e^{ρ+iθ}) + e^{−ρ} sin θ h(w, e^{ρ+iθ}).

The equations become

∂_{ρ}a − dh − ∗(∂_{θ}a − df) = 0
∂_{ρ}f − ∂_{θ}h − e^{2ρ} ∗ F_{a} = 0.

Up to a gauge transformation we can assume that the equations are in radial gauge, that is h ≡ 0 for ρ large enough. Thus, we get

∂_{ρ}a − ∗(∂_{θ}a − df) = 0  (6)

∂_{ρ}f − e^{2ρ} ∗ F_{a} = 0.  (7)

Notice that the equation in this form can be interpreted as the abelian ASD equation on T^{2} × S^{1} × R, where the metric on T^{2} has a conformal factor depending on ρ ∈ R, of the form e^{−ρ}g_{T^{2}}, where g_{T^{2}} is the standard metric on the flat torus. In fact the ∗ operator on p-forms rescales like ∗_{ρ} = (e^{−ρ})^{2−2p}∗.

We are considering finite energy solutions, that is, we impose the condition

∫_{−∞}^{∞} ∫_{0}^{2π} (∥∂_{ρ}a∥_{L^{2}(T^{2}),g_{T^{2}}}^{2} + e^{2ρ}∥F_{a}∥_{L^{2}(T^{2}),g_{T^{2}}}^{2} ) dθdρ < ∞.

Under a change of variables α = e^{ρ}a we rewrite equations (6) and (7) as

∂_{ρ}α = α + ∗∂_{θ}α − e^{ρ} ∗ df  (8)

∂_{ρ}f = e^{ρ} ∗ dα.  (9)

We can write α = u(w, ρ, θ)dx + v(w, ρ, θ)dy, where w = (x, y) are the coordinates on the torus T^{2}. We can expand u, v, and f in Fourier series in the variables θ, x, and y.

We get

u(w, θ, ρ) = ∑ u_{n}(ρ)_{lk} \frac{e^{inθ}}{(2π)^{1/2}} \frac{e^{ilx}}{(2π)^{1/2}} \frac{e^{iky}}{(2π)^{1/2}} ,

v(w, θ, ρ) = ∑ v_{n}(ρ)_{lk} \frac{e^{inθ}}{(2π)^{1/2}} \frac{e^{ilx}}{(2π)^{1/2}} \frac{e^{iky}}{(2π)^{1/2}} ,

f(w, θ, ρ) = ∑ f_{n}(ρ)_{lk} \frac{e^{inθ}}{(2π)^{1/2}} \frac{e^{ilx}}{(2π)^{1/2}} \frac{e^{iky}}{(2π)^{1/2}} .

The system of equations becomes the ODE

\frac{d}{dρ} ( \begin{matrix}u_{n}(ρ)_{lk} \\ v_{n}(ρ)_{lk} \\ f_{n}(ρ)_{lk} \end{matrix} ) = ( \begin{matrix}1 & −in & ike^{ρ} \\ in & 1 & −ile^{ρ} \\ ike^{ρ} & −ile^{ρ} & 0 \end{matrix} ) ( \begin{matrix}u_{n}(ρ)_{lk} \\ v_{n}(ρ)_{lk} \\ f_{n}(ρ)_{lk} \end{matrix} ) .  (10)

In the case of the asymptotics on the end T^{2}×[0, ∞)×R, we consider the same equations and restrict solutions to the domain θ ∈ [−π/2, π/2], in fact, we are not imposing any boundary conditions at θ = −π/2 and θ = π/2, other than the functions being smooth across θ = −π/2 and θ = π/2.

The assumption that A is a U(1) connection imposes the constraint on the coefficients

u_{n}(ρ)_{lk} = −\overline{u_{−n}(ρ)_{−l,−k}},
v_{n}(ρ)_{lk} = −\overline{v_{−n}(ρ)_{−l,−k}},
f_{n}(ρ)_{lk} = −\overline{f_{−n}(ρ)_{−l,−k}}.

A direct analysis of this system of ODE’s (cf. [9], §X) proves the following Proposition.

Proposition 3.4 The only finite energy solutions of the abelian ASD equation on T^{2} × R^{2} are flat connections on T^{2}, constant in the R^{2} directions. On the asymptotic end T^{2} × [r_{0}, ∞) × R all the finite energy solutions are of the form

\begin{aligned}a(w, ρ, θ) = & (u_{0} + ∑_{n≠0} u_{n}e^{−|n|ρ+inθ} \\ & + ∑_{n,l,k} lc_{nlk}e^{i(lx+ky+nθ)})dx+ \\ & (v_{0} + ∑_{n≠0} v_{n}e^{−|n|ρ+inθ} \\ & + ∑_{n,l,k} kc_{nlk}e^{i(lx+ky+nθ)})dy \\ f(w, ρ, θ) = & f_{0} + ∑_{n,l,k} nc_{nlk}e^{i(lx+ky+nθ)} \end{aligned}  (11)

with the coefficients that satisfy

∑_{n} |nu_{n}|^{2} + ∑_{n,l,k} |(n^{2} + l^{2} + k^{2})c_{nlk}|^{2} < ∞,

and

c_{nlk} = −\overline{c_{−n,−l,−k}} and u_{n} = −\overline{u_{−n}}.

In radial gauge, the limit in the ρ → ∞ direction is given by

a_{∞}(w, θ) = (u_{0} + ∑_{n,l,k} lc_{nlk}e^{i(lx+ky+nθ)})dx+
(v_{0} + ∑_{n,l,k} kc_{nlk}e^{i(lx+ky+nθ)})dy
f_{∞}(w, θ) =  f_{0} + ∑_{n,l,k} nc_{nlk}e^{i(lx+ky+nθ)}.

Proof.

It is easy to find a special family of solutions of the form

c_{nlk} ( \begin{matrix}l e^{ρ} \\ ke^{ρ} \\ n \end{matrix} ) ,

with −\overline{c_{−n,−l,−k}} = c_{nlk}. These correspond to the condition dα = 0 in the equation (9). In the original variables, that is multiplying the connection terms by e^{−ρ}, these are solutions of (6) and (7) constant in the ρ-direction.

We assume for the moment that (k, l) ≠ (0, 0). Following the standard methods of ODE theory, we use these solutions in order to reduce the system to a system of two equations. If we assume l ≠ 0, we can consider the matrix

Z = ( \begin{matrix}le^{ρ} & 0 & 0 \\ ke^{ρ} & 1 & 0 \\ n & 0 & 1 \end{matrix} ) .

The change of variables

( \begin{matrix}u \\ v \\ f \end{matrix} ) = Z ( \begin{matrix}ω \\ ν \\ ϕ \end{matrix} )

gives the new system of equations

\frac{d}{dρ} ( \begin{matrix}ω \\ ν \\ ϕ \end{matrix} ) = Z^{−1}A ( \begin{matrix}0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}ω \\ ν \\ ϕ \end{matrix} ) ,

where A is the matrix

A = ( \begin{matrix}1 & −in & 0 \\ in & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

This system consists of the equation

ω′ = \frac{−in}{l}e^{−ρ}ν + \frac{ik}{l} ϕ

and the two by two system

\frac{d}{dρ} ( \begin{matrix}lν \\ lϕ \end{matrix} ) = ( \begin{matrix}ink + l & −i(k^{2} + l^{2})e^{ρ} \\ i(n^{2}e^{−ρ} − l^{2}e^{ρ}) & −ink \end{matrix} ) ( \begin{matrix}ν \\ ϕ \end{matrix} ) .

Since we are interested in the large ρ >> 0 behavior of the system we can isolate a leading term and treat the rest of the system as a perturbation.

Let us define

L_{ρ} = ( \begin{matrix}0 & −i(k^{2} + l^{2})e^{ρ} \\ −il^{2}e^{ρ} & 0 \end{matrix} )

to be the leading term and

P_{ρ} = ( \begin{matrix}ink + l & 0 \\ in^{2}e^{−ρ} & −ink \end{matrix} )

to be the perturbation.

The unperturbed system has eigenvalues

λ_{±}(ρ) = ±ie^{ρ}(k^{2} + l^{2})^{1/2},

with eigenvectors

U_{±} = ( 1_{\frac{±l}{(k^{2}+l^{2})^{1/2}}} ) .

So, upon diagonalizing the matrix we obtain solutions exp(±ie^{ρ}(k^{2} +l^{2})^{1/2}), hence in the original system of coordinates we have solutions

\begin{aligned}ν(ρ) = & \frac{(k^{2}+l^{2})^{1/2}}{2l} c_{1} exp(ie^{ρ}(k^{2} + l^{2})^{1/2}) \\ & +\frac{(k^{2}+l^{2})^{1/2}}{2l} c_{2} exp(−ie^{ρ}(k^{2} + l^{2})^{1/2}), \end{aligned}  (12)

and

\begin{aligned}ϕ(ρ) = & −\frac{1}{2}c_{1} exp(ie^{ρ}(k^{2} + l^{2})^{1/2}) \\ & +\frac{1}{2}c_{2} exp(−ie^{ρ}(k^{2} + l^{2})^{1/2}). \end{aligned}  (13)

with the remaining equation that gives

ω(ρ) = ∫_{0}^{ρ} (\frac{−in(k^{2}+l^{2})^{1/2}}{l^{2}} (c_{1} exp(ie^{τ}) + c_{2} exp(−ie^{τ})) −
\frac{ik}{2l}e^{τ} (c_{1} exp(ie^{τ}) − c_{2} exp(−ie^{τ}))) dτ.

(14)

The perturbed system has eigenvalues

λ̃_{±}(ρ) = \frac{1 ± \sqrt{1 − 4(e^{2ρ}(k^{2} + l^{2}) − n^{2} − \frac{ink}{l})}}{2}.

The matrix can be diagonalized so that the solutions are of the form

exp (∫_{0}^{ρ} λ̃_{±}(τ)dτ) .

The long distance ρ >> 0 behavior of these solutions is given by the asymptotics e^{ρ±ie^{ρ}}. On the other hand the eigenvectors become asymptotically ρ-independent and approach the eigenvectors U_{±} of the unperturbed system. Thus, the asymptotic behavior of the solutions will be of the form

ν(ρ) ∼ \frac{(k^{2} + l^{2})^{1/2}}{2l} (c_{1}e^{ρ+ie^{ρ}} + c_{2}e^{ρ−ie^{ρ}})  (15)

and

ϕ(ρ) ∼ \frac{−1}{2} (c_{1}e^{ρ+ie^{ρ}} − c_{2}e^{ρ−ie^{ρ}})  (16)

The third variable is then obtained as ω(ρ) ∼ ∫_{0}^{ρ} (\frac{−in(k^{2}+l^{2})^{1/2}}{l^{2}} (c_{1} exp(ie^{τ}) + c_{2} exp(−ie^{τ})) −

\frac{ik}{2l}e^{τ} (c_{1} exp(ie^{τ}) − c_{2} exp(−ie^{τ}))) dτ.

(17)

Then the original solutions will be of the form

( \begin{matrix}u \\ v \\ f \end{matrix} ) = ( \begin{matrix}le^{ρ}ω(ρ) \\ ke^{ρ}ω(ρ) + ν(ρ) \\ nω(ρ) + ϕ(ρ) \end{matrix} ) .

The case with l = 0 and k ≠ 0 is analogous. In fact, in that case we can use a similar reduction by considering the matrix

Z = ( \begin{matrix}0 & 1 & 0 \\ ke^{ρ} & 0 & 0 \\ n & 0 & 1 \end{matrix} ) .

The change of variables

( \begin{matrix}u \\ v \\ f \end{matrix} ) = Z ( \begin{matrix}ω \\ ν \\ ϕ \end{matrix} )

gives the new system of equations

\frac{d}{dρ} ( \begin{matrix}ω \\ ν \\ ϕ \end{matrix} ) = Z^{−1}A ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ) ( \begin{matrix}ω \\ ν \\ ϕ \end{matrix} ) ,

which is of the form

ω′ = \frac{in}{k}e^{−ρ}ν

and the remaining two by two system

\frac{d}{dρ} ( \begin{matrix}ν \\ ϕ \end{matrix} ) = ( \begin{matrix}1 & ike^{ρ} \\ \frac{−in^{2}}{k} e^{−ρ} + ike^{ρ} & 0 \end{matrix} ) ( \begin{matrix}ν \\ ϕ \end{matrix} ) .

Again we can isolate a leading term

L_{ρ} = ( \begin{matrix}0 & ike^{ρ} \\ ike^{ρ} & 0 \end{matrix} )

and a perturbation

P_{ρ} = ( \begin{matrix}1 & 0 \\ \frac{−in^{2}}{k} e^{−ρ} & 0 \end{matrix} ) .

The unperturbed system has eigenvalues λ_{±} = ±ike^{ρ} and ρ-independent eigenvectors

U_{±} = ( \begin{matrix}1 \\ ±1 \end{matrix} ) .

The perturbed system has eigenvalues

λ̃_{±} = \frac{1 ± \sqrt{1 − 4(k^{2}e^{2ρ} − n^{2})}}{2}.

The asymptotics of the solutions is exp(ρ ± ike^{ρ}), with eigenvectors that asymptotically approach the ρ-independent eigenvectors U_{±}. Thus, we obtain asymptotics

ω(ρ) ∼ \frac{in}{2k} ∫_{0}^{ρ}(c_{1}e^{+ike^{τ}} + c_{2}e^{−ike^{τ}})dτ,

ν(ρ) ∼ \frac{1}{2}(c_{1}e^{ρ+ike^{ρ}} + c_{2}e^{ρ−ike^{ρ}})

and

ϕ(ρ) ∼ \frac{1}{2}(c_{1}e^{ρ+ike^{ρ}} − c_{2}e^{ρ−ike^{ρ}}).

Among these families of solutions, the only elements that satisfy the finite energy condition are written in the original variables as solutions of (6) and (7) of the form

\begin{aligned}a(w, ρ, θ) = & (u_{0} + ∑_{n,l,k} lc_{nlk}e^{i(lx+ky+nθ)})dx+ \\ & (v_{0} + ∑_{n,l,k} kc_{nlk}e^{i(lx+ky+nθ)})dy \\ f(w, ρ, θ) = & f_{0} + ∑_{n,l,k} nc_{nlk}e^{i(lx+ky+nθ)}. \end{aligned}  (18)

The remaining solutions satisfy (k, l) = (0, 0). These satisfy the conditions df = 0 and dα = 0 in the equations (8) and (9). These are solutions of the form

( \begin{matrix}c_{1}(n, 0, 0)e^{(1+n)t} − ic_{2}(n, 0, 0)e^{(1−n)t} \\ c_{1}(n, 0, 0)e^{(1+n)t} + ic_{2}(n, 0, 0)e^{(1−n)t} \\ c_{3}(n, 0, 0) \end{matrix} )

to the system with constant coefficients with matrix

A = ( \begin{matrix}1 & −in & 0 \\ in & 1 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

This gives solutions of the original equations (6) and (7) of the form

\begin{aligned}a(w, ρ, θ) = & (u_{0} + ∑_{n≠0} u_{n}e^{−|n|ρ+inθ}) dx \\ & + (v_{0} + ∑_{n≠0} v_{n}e^{−|n|ρ+inθ}) dy \\ f(w, ρ, θ) = & f_{0} \end{aligned}  (19)

This proves the Proposition. In fact, we can write the function

f_{∞} − f_{0} = −i\frac{d}{dθ} ∑_{n,l,k} c_{nlk}e^{i(lx+ky+nθ)},

so that, if we define

γ(θ) = ∑_{l,k} −ic_{nlk}e^{i(lx+ky+nθ)},

the flat connection a_{∞} can be written as

a_{∞} − (u_{0}dx + v_{0}dy) = dγ(θ).

This means that the path of asymptotic flat connections in the covering χ_{0}(T^{2}, ν(K)) of the character variety χ(T^{2}) can be written as

a_{∞}(θ) = λ(θ) · a_{0},

where λ : T^{2} × [0, π] → U(1) satisfies

λ^{−1}(θ)d_{T^{2}}λ(θ) = d_{T^{2}}γ(θ).

⋄ Thus we have obtained the following.

Corollary 3.5 A finite energy solution of the abelian ASD equation on an asymptotic end of the form T^{2} × [0, ∞) × R decays exponentially along the radial direction. The asymptotic values for different angles θ are all within the same gauge class of flat connections on T^{2}. Thus, we have decay to a point a_{∞} = [a_{∞}(θ)].

Remark 3.6 Up to a global change of gauge, the only solutions of the form (11) that extend to all of ν(K) × R are flat connections on T^{2} constant in the s and t directions.

In the light of Remark 3.6, the explicit analysis of the asymptotics (11) seems a somewhat useless complication. However, the analysis above will be crucial in the following section, in order to describe, with a perturbative analysis, the asymptotic behavior of finite energy solutions of the Seiberg-Witten equations on V × R with an end T^{2} × [0, ∞) × R.

3.2 Asymptotics of monopoles on V × R

We now analyze the asymptotics of solutions of the four-dimensional Seiberg-Witten equations on the end T^{2} × [r_{0}, ∞) × R of the manifold V × R, where V is the knot complement in V, endowed with an infinite cylindrical end.

We write the connection A = a(w, s, t) + f(w, s, t)ds + h(w, s, t)dt, and the spinor section Ψ = (α, β) with α(s, t) ∈ Λ^{0,0}(T^{2}) and β(s, t) ∈ Λ^{0,1}(T^{2}).

The Seiberg-Witten equations can be written in the form

∂_{t}a − dh + ∗(∂_{s}a − df) = ∗i(ᾱβ + αβ̄)

∂_{t}f − ∂_{s}h + ∗F_{a} = \frac{i}{2}(|α|^{2} − |β|^{2}),

for the curvature equation, and

∂_{t}α + hα + i∂_{s}α + ifα + ¯∂_{a}^{∗}β = 0
∂_{t}β + hβ − i∂_{s}β − ifβ + ¯∂_{a}α = 0,

for the Dirac equation.

As before, we introduce the variables w = (x, y) ∈ T^{2}, and z = s + it, z = e^{ρ+iθ} on [r_{0}, ∞) × R, and the change of coordinates

a(w, ρ, θ) = a(w, e^{ρ+iθ})
f(w, ρ, θ) = e^{−ρ} cos θ h(w, e^{ρ+iθ}) − e^{−ρ} sin θ f(w, e^{ρ+iθ})
h(w, ρ, θ) = e^{−ρ} cos θ f(w, e^{ρ+iθ}) + e^{−ρ} sin θ h(w, e^{ρ+iθ})
α(w, ρ, θ) = α(w, e^{ρ+iθ})
β(w, ρ, θ) = β(w, e^{ρ+iθ}).

(20)

The Seiberg-Witten equations in radial gauge (i.e. with h ≡ 0 for large ρ) are then written in the form

\begin{matrix}∂_{ρ}a = ∗(∂_{θ}a − df + i(ᾱβ + αβ̄)) \\ ∂_{ρ}f = e^{2ρ} ∗ (F_{a} + \frac{i}{2}(|α|^{2} − |β|^{2})ω) \\ ∂_{ρ}α = i(∂_{θ}α + fα + e^{ρ+iθ} ¯∂_{a}^{∗}β) \\ ∂_{ρ}β = −i(∂_{θ}β + fβ + e^{ρ−iθ} ¯∂_{a}α). \end{matrix}  (21)

In order to study the asymptotic behavior of these solutions, we can again use Fourier transform. We first analyze the asymptotic behavior of a linear system and then introduce the non-linear terms in a sequence of successive approximations [9]. In order to simplify the expression of the quadratic terms in (21), we use the notation

A · B = ∑(ab)_{nlk} \frac{e^{i(nθ+lx+ky)}}{(2π)^{3/2}} ,

with the coefficients given by

(ab)_{nlk} = ∑ \frac{1}{(2π)^{3/2}} a_{ijh}b_{n−i l−j k−h},

for

A = ∑ a_{nlk} \frac{e^{i(nθ+lx+ky)}}{(2π)^{3/2}}

and

B = ∑ b_{nlk} \frac{e^{i(nθ+lx+ky)}}{(2π)^{3/2}} .

With this notation, the equation (21) becomes

\begin{aligned}u′_{nlk} = & −inv_{nlk} + ikf_{nlk} + 2iRe(ᾱβ)_{nlk} \\ v′_{nlk} = & inu_{nlk} − ilf_{nlk} + 2iIm(ᾱβ)_{nlk} \\ f′_{nlk} = & ike^{2ρ}u_{nlk} − ile^{2ρ}v_{nlk} + e^{2ρ}(ᾱα)_{nlk} − (β̄β)_{nlk} \\ α′_{nlk} = & −nα_{nlk} + ie^{ρ} \frac{1}{2}(il − k)β̄_{−n−1 −l −k}+ \\ & ie^{ρ} \frac{1}{2}(aβ̄)_{nlk} + i(fα)_{nlk} \\ β′_{nlk} = & nβ_{nlk} − ie^{ρ}(il − k)α_{n+1 lk} \\ & −ie^{ρ}(aα)_{nlk} − i(fβ)_{nlk}. \end{aligned}  (22)

Consider the linear system of equations

\begin{matrix}∂_{ρ}a = ∗(∂_{θ}a − df) \\ ∂_{ρ}f = e^{2ρ} ∗ F_{a} \\ ∂_{ρ}α = i(∂_{θ}α + e^{ρ+iθ} ¯∂^{∗}β) \\ ∂_{ρ}β = −i(∂_{θ}β + e^{ρ−iθ} ¯∂α). \end{matrix}  (23)

This is the uncoupled system of the abelian ASD equation and the linear Dirac equation. We analyzed the first two equations in the previous section.

In the Dirac equations, since again we are interested in the large ρ >> 0 behavior of the finite energy solutions, we can isolate a leading term and a perturbation.

The leading term gives a system of the form

\begin{matrix}α′_{nlk} = ie^{ρ} \frac{1}{2}(il − k)β̄_{−n−1 −l −k} \\ β′_{nlk} = −ie^{ρ}(il − k)α_{n+1 lk}, \end{matrix}  (24)

Whereas we identify perturbation all the terms of the form

( \begin{matrix}−n & 0 \\ 0 & n \end{matrix} ) ( \begin{matrix}α_{nlk} \\ β_{nlk} \end{matrix} ) .  (25)

We analyze solutions to the unperturbed system (24). If we introduce the notation α_{nlk} = η_{nlk} + iξ_{nlk} and β_{nlk} = p_{nlk} + iq_{nlk}, the system (24) uncouples in the independent systems of eight equations:

\begin{matrix}p′_{nlk} = e^{ρ}(lη_{n+1 lk} − kξ_{n+1 lk}) \\ q′_{nlk} = e^{ρ}(lξ_{n+1 lk} + kη_{n+1 lk}) \\ η′_{n+1 lk} = \frac{e^{ρ}}{2} (−lp_{−n−2 −l −k} − kq_{−n−2 −l −k}) \\ ξ′_{n+1 lk} = \frac{e^{ρ}}{2} (lp_{−n−2 −l −k} − kq_{−n−2 −l −k}) \\ p′_{−n−2 −l −k} = e^{ρ}(−lη_{−n−1 −l −k} + kξ_{−n−1 −l −k}) \\ q′_{−n−2 −l −k} = e^{ρ}(−lη_{−n−1 −l −k} − kξ_{−n−1 −l −k}) \\ η′_{−n−1 −l −k} = \frac{e^{ρ}}{2} (lp_{nlk} + kq_{nlk}) \\ ξ′_{−n−1 −l −k} = \frac{e^{ρ}}{2} (−lp_{nlk} + kq_{nlk}) \end{matrix}  (26)

Consider the matrices

A = ( \begin{matrix}l & −k \\ l & k \end{matrix} )

and

B = \frac{1}{2} ( \begin{matrix}−l & −k \\ l & −k \end{matrix} ) .

Up to a reparametrization in the variable τ = e^{ρ}, we obtain the autonomous linear system with the 8 × 8 matrix

M = ( \begin{matrix}0 & A & 0 & 0 \\ 0 & 0 & B & 0 \\ 0 & 0 & 0 & −A \\ −B & 0 & 0 & 0 \end{matrix} ) .

If we write η_{0} = (k^{2} + 6kl + l^{2})^{1/2}, η_{1} = k − l, and η_{2} = k + l, then the eigenvalues of M are of the form ±λ_{i}^{nlk}, for i = 1, … 4, with

\begin{matrix}λ_{1}^{nlk} = \frac{1}{2}(η_{2}^{2} + η_{1}η_{0})^{1/2} & λ_{2}^{nlk} = \frac{1}{2}(η_{2}^{2} − η_{1}η_{0})^{1/2} \\ λ_{3}^{nlk} = \frac{1}{2}(−η_{2}^{2} − η_{1}η_{0})^{1/2} & λ_{4}^{nlk} = \frac{1}{2}(−η_{2}^{2} + η_{1}η_{0})^{1/2}. \end{matrix}  (27)

It is easy to see that, in the basis U_{i}^{±}(nlk) of eigenvectors, the solutions corresponding to λ_{1}^{nlk} and λ_{2}^{nlk} are saddles for all non-trivial (n, l, k) and solutions corresponding to λ_{3}^{nlk} and λ_{4}^{nlk} are centers for all non-trivial (n, l, k). Thus, the finite energy solutions will be of the form

( \begin{matrix}p_{nlk} \\ q_{nlk} \\ η_{n+1 lk} \\ ξ_{n+1 lk} \\ p_{−n−2 −l −k} \\ q_{−n−2 −l −k} \\ η_{−n−1 −l −k} \\ ξ_{−n−1 −l −k} \end{matrix} ) = ∑_{i=1}^{2} c_{i}^{nlk} exp(−|λ_{i}^{nlk}|τ)U_{i}^{−}(nlk),  (28)

for τ = e^{ρ}.

If we add the perturbation terms (25) we can proceed as in the case of the ASD equation and observe that, for large ρ → ∞ the eigenvalues and eigenvectors of the perturbed system converge to the eigenvalues and eigenvectors ±λ_{i}^{nlk} and U_{i}^{±}(nlk) described above.

In the variables α_{nlk} = η_{nlk} +iξ_{nlk} and β_{nlk} = p_{nlk} +iq_{nlk}, the perturbed system can be written as the non-autonomous linear system with the 8 × 8 matrix

M̃ = ( \begin{matrix}nI & e^{ρ}A & 0 & 0 \\ 0 & −(n + 1)I & e^{ρ}B & 0 \\ 0 & 0 & −(n + 2)I & −e^{ρ}A \\ −e^{ρ}B & 0 & 0 & (n + 1)I \end{matrix} ) ,  (29)

where I is the 2 × 2 identity matrix.

The eigenvalues have a term independent of ρ which reduces to the diagonal entries of the matrix M̃ as ρ → −∞, and terms containing e^{ρ}, of which the highest order term reduces to the eigenvalues of the matrix M. Thus, the rate of decay as ρ → ∞ in the directions corresponding to eigenvalues of negative real part is at least given by exp(−nρ) and at most by exp(−C_{nlk}e^{ρ}), for some positive constants C_{nlk} > 0, or by mixed terms exp(nρ − C_{nlk}e^{ρ}).

With this analysis in place, we know the asymptotic behavior as ρ → ∞ of the finite energy solutions of the linear system. Now we want to analyze the asymptotic behavior of solutions of the original non-linear equations.

Suppose given a finite energy solution Ξ_{0} = (u_{0}, v_{0}, f_{0}, α_{0}, β_{0}) of the linear system. Consider a perturbed linear system with matrix L_{Ξ_{0}} as follows:

\begin{aligned}u′_{nlk} = & −inv_{nlk} + ikf_{nlk} + 2iRe(ᾱ_{0}β)_{nlk} + 2iRe(ᾱβ_{0})_{nlk} \\ v′_{nlk} = & inu_{nlk} − ilf_{nlk} + 2iIm(ᾱ_{0}β)_{nlk} + 2iIm(ᾱβ_{0})_{nlk} \\ f′_{nlk} = & ike^{2ρ}u_{nlk} − ile^{2ρ}v_{nlk} \\ & +2e^{2ρ}(Re(ᾱ_{0}α)_{nlk} − (β̄_{0}β)_{nlk}) \\ α′_{nlk} = & −nα_{nlk} + ie^{ρ} \frac{1}{2}(il − k)β̄_{−n−1 −l −k} \\ & +ie^{ρ} \frac{1}{2}(a_{0}β̄)_{nlk} + ie^{ρ} \frac{1}{2}(aβ̄_{0})_{nlk} + i(f_{0}α)_{nlk} + i(fα_{0}) \\ β′_{nlk} = & nβ_{nlk} − ie^{ρ}(il − k)α_{n+1 lk} − ie^{ρ}(a_{0}α)_{nlk} − ie^{ρ}(aα_{0})_{nlk} \\ & −i(f_{0}β)_{nlk} − i(fβ_{0}). \end{aligned}  (30)

Similarly, we can define inductively solutions Ξ_{ν+1} satisfying the linear system of equations

Ξ′_{ν+1} = L_{Ξ_{ν}}Ξ_{ν+1},

with initial condition Ξ_{ν+1}(0) = Ξ_{ν}(0).

Proposition 3.7 All the solutions Ξ_{ν+1} are finite energy. Moreover, we have

Ξ_{ν} → Ξ uniformly on compact sets along with all derivative, where Ξ is a solution of the original non-linear system (21)

Ξ′ = L_{Ξ}Ξ.

The solution Ξ has the same asymptotic behavior as ρ → ∞ as Ξ_{0}, up to terms that decay faster as ρ → ∞.

Proof. Inductively, the system of equations satisfied by Ξ_{ν+1} can be separated into a leading term and a perturbation. The rate of decay of the solutions to the system with the leading term only will determine the asymptotics of Ξ_{ν+1}.

In the case of the system (30), we have the coefficients of Ξ_{0} decaying at least like e^{−nρ}. Thus, for large ρ, the leading terms in the system (30) will be the same as the leading terms in the original linear system.

Inductively, at the ν-th stage, we have the original linear system

\begin{aligned}u′_{nlk} = & −inv_{nlk} + ikf_{nlk} \\ & 2iRe(ᾱ_{ν}β)_{nlk} + 2iRe(ᾱβ_{ν})_{nlk} \\ v′_{nlk} = & inu_{nlk} − ilf_{nlk} \\ & 2iIm(ᾱ_{ν}β)_{nlk} + 2iIm(ᾱβ_{ν})_{nlk} \\ f′_{nlk} = & ike^{2ρ}u_{nlk} − ile^{2ρ}v_{nlk} \\ & 2e^{2ρ}(Re(ᾱ_{ν}α)_{nlk} − (β̄_{ν}β)_{nlk}) \\ α′_{nlk} = & −nα_{nlk} + ie^{ρ} \frac{1}{2}(il − k)β̄_{−n−1 −l −k} \\ & ie^{ρ} \frac{1}{2}(a_{ν}β̄)_{nlk} + ie^{ρ} \frac{1}{2}(aβ̄_{ν})_{nlk} + i(f_{ν}α)_{nlk} + i(fα_{ν}) \\ β′_{nlk} = & nβ_{nlk} − ie^{ρ}(il − k)α_{n+1 lk} \\ & −ie^{ρ}(a_{ν}α)_{nlk} − ie^{ρ}(aα_{ν})_{nlk} − i(f_{ν}β)_{nlk} − i(fβ_{ν}). \end{aligned}  (31)

Under the inductive hypothesis that the coefficients of the solutions Ξ_{ν} decay at a rate at least e^{−nρ} as ρ → ∞, we see that the leading terms in the system (31) are the same as in the original unperturbed system. The eigenvalues ±λ̃_{i}^{nkl}(ρ) of the perturbed system (31) and the corresponding eigenvectors Ũ_{i}^{±}(ρ) converge asymptotically to the ρ-independent eigenvalues and eigenvectors of the matrix M. Thus, the solutions Ξ_{ν} are of finite energy and maintain the same rate of decay as the original solution Ξ_{0}. ⋄

3.3 Uniform convergence

Notice that, in the particular case where the pointwise convergence proved in Lemma 3.1 extends uniformly to an infinite strip T^{2} ×[−r_{0}, r_{0}]×R, in the sense specified below, then we have uniform convergence of the (A_{r}, Ψ_{r}) in the smooth topology on the infinite strip to a flat connection. In particular, if we consider the geometric limits of solutions on (1) on Y (r) as r → ∞, as analysed in [4], we see that, in this special case, the asymptotic values (A_{r}(±∞), ψ_{r}(±∞)) break through the same flat connection on T^{2}, when r → ∞.

By the results of the previous section, on any fixed compact set K in T^{2} × [−r, r] × R, or in ν(K) × R, we have

∥F_{A_{r}}∥_{L^{2}(K)} → 0.

Proposition 3.8 Suppose given a family of finite energy solutions Ξ_{r} = (A_{r}, Ψ_{r}), for all r ≥ r_{0}. Suppose there exists a strip T^{2} × [r_{0}, r_{1}] × R in T^{2} × [−r, r] × R such that we have

∥F_{A_{r}}∥_{L^{2}(T^{2}×[r_{0},r_{1}]×R)} → 0  (32)

as r → ∞. Then there exists a flat connection a_{T^{2}} on T^{2} such that (up to gauge transformations) the sequence Ξ_{r} converges in the L_{2}^{2}-topology on T^{2} × [r_{0}, r_{1}] × R to a_{T^{2}}. In particular this implies that, under the splitting of the critical points M_{Y (r)} = M_{V} #_{χ_{0}(T^{2})}χ(ν(K)), for large enough r, the asymptotic values (A_{r}(±∞), ψ_{r}(±∞)) can be written as the gluing

(A_{r}(±∞), ψ_{r}(±∞)) = (A′(±∞), ψ′(±∞))#_{a_{T^{2}}} (λ^{±}a_{T^{2}}, 0),

that is, they split through the same asymptotic flat connection on T^{2}. Moreover, suppose that (A′, Ψ′) and (A, 0) are the limits on compact sets on V × R and ν(K) × R respectively, with asymptotic values [a_{V}] ∈ χ_{0}(T^{2}, V) and [a_{ν}] ∈ χ_{0}(T^{2}, ν(K)), as described in the previous sections. Then, in this particular case, there are gauge transformations λ′ and λ on V and ν(K) respectively such that we have

λ′a_{V} = a_{T^{2}} = λa_{ν}.

Proof. By Uhlenbeck weak convergence [8] [18], and the result of Proposition 8.3 of [8], the condition

∥F_{A_{r}}∥_{L^{2}(T^{2}×[r_{0},r_{1}]×R)} → 0

implies that there exists a flat connection A on T^{2} × [r_{0}, r_{1}] × R such that, up to gauge transformations λ_{r} in L_{3}^{2} and up to passing to a subsequence, we have

∥λ_{r}A_{r} − A∥_{L_{2}^{2}(T^{2}×[r_{0},r_{1}]×R)} ≤ C∥F_{A_{r}}∥_{L^{2}(T^{2}×[r_{0},r_{1}]×R)} → 0.

We can now show that we can further gauge transform the connections A_{k} so that this Uhlenbeck limit on T^{2} × [r_{0}, r_{1}] × R is a flat connection on T^{2}, constant in the t ∈ R and s ∈ [r_{0}, r_{1}] directions.

Claim: Let A be an L_{2,loc}^{2} flat connection on T^{2} × [r_{0}, r_{1}] × R. Then, there exist and L_{3,loc}^{2} gauge transformation λ on T^{2} × [r_{0}, r_{1}] × R such that λA is a flat connection a_{T^{2}} on T^{2}, constant in the t and s directions.

proof of Claim: We can write A in the form A = a_{T^{2}} + if dt + ih ds, where a_{T^{2}} is a connection on T^{2}, and f(x, y, t, s) and h(x, y, t, s) are real valued functions. Then the equation F_{A} = 0 becomes

\begin{matrix}F_{a_{T}^{2}} = 0 \\ ∂_{t}a_{T^{2}} = d_{T^{2}}if \\ ∂_{s}a_{T^{2}} = d_{T^{2}}ih \\ ∂_{t}h = ∂_{s}f. \end{matrix}  (33)

Define the gauge transformation

λ = exp (−i ∫_{r_{0}}^{s} h(x, y, t, σ)dσ − i ∫_{0}^{t} f(x, y, τ, r_{0})dτ) .

This satisfies

λ^{−1}dλ = −ih(x, y, t, s)ds − if(x, y, t, r_{0})dt
+(−i ∫_{r_{0}}^{s} ∂_{t}h(x, y, t, σ)dσ)dt
−id_{T^{2}} (∫_{r_{0}}^{s} h(x, y, t, σ)dσ + ∫_{0}^{t} f(x, y, τ, r_{0})dτ)
= −ih(x, y, t, s)ds − if(x, y, t, s)dt
−id_{T^{2}} (∫_{r_{0}}^{s} h(x, y, t, σ)dσ + ∫_{0}^{t} f(x, y, τ, r_{0})dτ) ,

where the equality follows from the last equation of (33). The gauge transformed connection λA is of the form

λA = a_{T^{2}} − id_{T^{2}} (∫_{r_{0}}^{s} h(x, y, t, σ)dσ + ∫_{0}^{t} f(x, y, τ, r_{0})dτ) = ã_{T^{2}}.

This satisfies the equations

Fã_{T^{2}} = 0
∂_{t}ã_{T^{2}} = ∂_{t}a_{T^{2}} − id_{T^{2}}(f(x, y, t, r_{0}) + ∫_{r_{0}}^{s} ∂_{t}h(x, y, t, σ)dσ) =
= ∂_{t}a_{T^{2}} − id_{T^{2}}f = 0
∂_{s}ã_{T^{2}} = ∂_{s}a_{T^{2}} − id_{T^{2}}h = 0.

This completes the proof of the Claim.

The asymptotic limits a_{∞}^{±} of (A′(±∞), ψ′(±∞)) then satisfy

[a_{∞}^{+}] = [a_{∞}^{−}] = [a_{T^{2}}]

in the universal cover χ_{0}(T^{2}, Y) of χ(T^{2}). This completes the proof of the Proposition. ⋄

Notice, however, that in general the uniform convergence of Proposition 3.8 should not be expected. In fact, we have seen that the estimate (3) and Lemma 2.4 give

∥F_{A_{r}(t)}∥_{L^{2}(T^{2}×[r_{0},r_{1}]×\{t\})}^{2} ≤ Ce^{− \frac{c|t|}{r}} .

This allows for non-uniformity of the convergence: there may be sequences T_{r} → ∞ for which (32) is violated, and an estimate like

∥F_{A_{r}(t)}∥_{L^{2}(T^{2}×[r_{0},r_{1}]×{t})}^{2} ≥ c > 0

holds for all |t| ∈ [T_{r} − ℓ/2, T_{r} + ℓ/2], for some ℓ > 0 independent of r ≥ r_{0}. This is clearly compatible with the constraint (3).

In the next subsection we analyze the geometric limits of monopoles (A_{r}, Ψ_{r}) on the domains T^{2} ×[−r, r]×[T_{r}, ∞) and T^{2} ×[−r, r]×(−∞, −T_{r}]. This is where we encounter the non-uniform convergence which is not detected by limits on compact sets.

3.4 Holomorphic disks in χ(T^{2})

Consider the Seiberg-Witten equations in on T^{2} × [−r, r] × R. Let (A_{r}, Ψ_{r}) be a family of solutions, where we write the connections as

A_{r} = a_{r}(w, s, t) + f_{r}(w, s, t)ds + h_{r}(w, s, t)dt

and the spinors as Ψ_{r} = (α_{r}(w, s, t), β_{r}(w, s, t)).

Recall again that gauge equivalent solutions (A_{r}(t), ψ_{r}(t)), in a temporal gauge in the t ∈ R direction, decay exponentially at the asymptotic ends t → ±∞ to critical points (A_{r}(±∞), ψ_{r}(±∞)) of the Chern-Simons-Dirac functional.

Suppose given a fixed compact set T^{2}×[r_{0}, r_{1}] in Y (r). Choose a sequence T_{r} → ∞, for which the estimate (3) gives

∥a_{r}(w, s, ±T_{r}) − a^{±}(w)∥^{2} + ∥f_{r}(w, s, ±T_{r}) − f^{±}(w, s)∥^{2} ≤ Ce^{−cr}, (34)

in the L_{1}^{2}-norm on T^{2}×[r_{0}, r_{1}], where a^{±} are the limits of (A_{r}(±∞), ψ_{r}(±∞)) as r → ∞ on T^{2} × [r_{0}, r_{1}]. In the first part [4] of this work we showed that, in the gluing of solutions to the 3-dimensional Seiberg-Witten equations (1), we are only interested in critical points (A_{r}(±∞), ψ_{r}(±∞)) of the Chern-Simons-Dirac functional that break through smooth points a^{±} ≠ ϑ in the character variety χ(T^{2}), as r → ∞. Thus, we can assume that the elements f^{±}(w, s) are exponentially small on T^{2} × [r_{0}, r_{1}], for large enough r_{0} > 0,

∥f^{±}(w, s)∥_{L^{2}(T^{2}×{s})}^{2} ≤ Ce^{−δs}.

This exponential estimate follows from the analysis of the asymptotics of monopoles on a 3-manifold with an end modeled on T^{2} × [0, ∞), as in [4].

This simple observation implies that we should expect more complicated geometric limits than the limits on compact sets analyzed in the previous section. In fact, for T_{r} growing sufficiently fast, we have convergence to the limits a^{±} on T^{2} × [r_{0}, r_{1}] × {t}, with |t| ≥ T_{r}. Thus, whenever a^{+} ≠ a^{−}, we must have geometric limits which are not just constant flat connections on T^{2}.

As discussed previously, in polar coordinates z = s + it and z = e^{ρ+iθ}, with

a_{r}(w, ρ, θ) = a_{r}(w, e^{ρ+iθ})
f_{r}(w, ρ, θ) = e^{−ρ} cos θ h_{r}(w, e^{ρ+iθ}) − e^{−ρ} sin θ f_{r}(w, e^{ρ+iθ})
h_{r}(w, ρ, θ) = e^{−ρ} cos θ f_{r}(w, e^{ρ+iθ}) + e^{−ρ} sin θ h_{r}(w, e^{ρ+iθ}),

the curvature equation takes the form

∂_{ρ}a_{r} − dh_{r} = ∗(∂_{θ}a_{r} − df_{r} + i(ᾱ_{r}β_{r} + α_{r}β̄_{r}))

∂_{ρ}f_{r} − ∂_{θ}h_{r} = e^{2ρ} ∗ (F_{a_{r}} + \frac{i}{2}(|α_{r}|^{2} − |β_{r}|^{2})ω).

We are now interested in studying the convergence away from arbitrarily large compact sets. That is, we are interested in the convergence on regions of the form

\begin{aligned}Ω̃(r) = & T^{2} × [0, r] × [T_{r}, ∞) \\ & ∪T^{2} × [0, r) × (−∞, −T_{r}] \\ & ∪T^{2} × \{ρ ∈ [log T_{r}, ∞) θ ∈ [−π/2, π/2]\} \end{aligned} .

We choose a sequence T (r) such that, as in (34), the finite energy solutions (A_{r}, Ψ_{r}) on Y (r) × R are exponentially close to the asymptotic values a^{±} on

T^{2} × [−r_{0}, r_{0}] × ([T (r), ∞) ∪ (−∞, −T (r)].

We write T (r) = e^{T_{r}}, with T_{r} → ∞ as r → ∞.

In order to study the convergence on the cylinder T^{2} × [−r, r] × R, we introduce a suitable rescaling of the coordinates on T^{2} × R^{2}. In polar coordinates, we introduce the translation ρ ↦ ρ − T_{r}. In these new coordinates, the second part of the curvature equation becomes

∂_{ρ}f_{r} − ∂_{θ}h_{r} − e^{2T_{r}}e^{2ρ} ∗ (F_{a_{r}} + \frac{i}{2}(|α_{r}|^{2} − |β_{r}|^{2})ω) = 0.

Consider the unit disk D = {ρ ≤ 0} in the new coordinates. By the previous analysis of the convergence on compact sets we obtain smooth convergence of a subsequence to a solution of

∂_{ρ}a − ∗(∂_{θ}a − df) = 0
F_{a} = 0.

Up to a gauge transformation, this is a holomorphic map

a : D → χ_{0}(T^{2}, Y) = C.

Namely, a(ρ, θ) is the flat connection on T^{2} satisfying

a(ρ, θ)(w) = a(w, ρ, θ).

With a slight abuse of notation we identify a(ρ, θ) with the class in χ_{0}(T^{2}, Y) obtained modulo homotopically trivial gauge transformations. The map a has the property that it maps the center of the disk, ρ → −∞, to the flat connection a_{∞}, which is the limit on compact sets of the original sequence (A_{r}, Ψ_{r}). Moreover, by the finite energy condition, the map a is holomorphic up to the boundary, and the restriction to the boundary

a : S^{1} = ∂D = {ρ = 0} → χ_{0}(T^{2}, Y) = C

gives a curve in χ_{0}(T^{2}, Y). By our choice of T_{r}, this curve will satisfy a(0, π/2) = a^{+} and a(0, −π/2) = a^{−}.

Thus, we have obtained that the convergence on the T^{2} × [−r, r] × R part of Y (r) × R, away from compact sets, produces a holomorphic map. Now we want to describe the corresponding non-uniform convergence, away from compact sets, on the sub-domains of Y (r) × R that correspond to the knot complement and to the tubular neighborhood of the knot. We will use a similar rescaling of the coordinates.

Let V_{r} be the knot complement V with a cylinder of length r,

V_{r} = V ∪_{T^{2}×{0}} T^{2} × [0, r].

On the manifold V_{r} × R consider the domain

Ω̃(r) = V_{r} × ([T_{r}, ∞) ∪ (−∞, −T_{r}])
∪T^{2} × {ρ ∈ [log T_{r}, ∞) θ ∈ [−π/2, π/2]},

where we use the polar coordinates (ρ, θ) introduced before. On this domain consider the change of coordinates

ρ ↦ ρ − T_{r}

in the ρ ∈ R direction, and the corresponding change of coordinates

|t| ↦ e^{−T_{r}}|t|

in the t ∈ R direction, where the coordinates ρ and t are related by z = s+it and z = e^{ρ+iθ}, as before, with (s, t) ∈ [0, ∞) × R.

This produces the rescaled domain

V_{r} × ([T_{r}e^{−T_{r}}, ∞) ∪ (−∞, −T_{r}e^{−T_{r}}])
∪T^{2} × {ρ ∈ [log(T_{r}) − T_{r}, ∞) θ ∈ [−π/2, π/2]}.

Figure 1: The domain Ω̃(r) and its rescaling

Consider the domains

Ω(r) = V_{r} × ([−1, −T_{r}e^{−T_{r}}] ∪ [T_{r}e^{−T_{r}}, 1])
∪T^{2} × {ρ ∈ (−T_{r} + log(T_{r}), 0] θ ∈ [−π/2, π/2]},

and

Ω_{1}(r) = V_{r} × ([−1, −T_{r}e^{−T_{r}}] ∪ [T_{r}e^{−T_{r}}, 1]) ⊂ Ω(r),
Ω_{2}(r) = T^{2} × {ρ ∈ (−T_{r} + log(T_{r}), 0] θ ∈ [−π/2, π/2]} ⊂ Ω(r).

The finite energy solutions (A_{r}, Ψ_{r}) of the SW equations satisfy

e^{−T_{r}} \frac{∂}{∂_{t}} A_{r}(t) = − ∗ F_{A_{r}(t)} + σ(ψ_{r}(t), ψ_{r}(t))

e^{−T_{r}} \frac{∂}{∂_{t}} ψ_{r}(t) = −∂_{A_{r}(t)}ψ_{r}(t)

on Ω_{1}(r).

The finite energy condition ensures that the left hand side converges to zero uniformly on any fixed compact set. Thus, the family (A_{r}, Ψ_{r}), under the present change of coordinates, converges smoothly on compact sets in to a path (A(t), ψ(t)) of finite energy solutions in the domain V × R^{∗}, where R^{∗} = R\{0}, where V is endowed with an infinite cylindrical end T^{2}×[0, ∞). That is, we obtain a path (A(t), ψ(t)) of elements satisfying

∗ F_{A(t)} + σ(ψ(t), ψ(t)) = 0
∂_{A(t)}ψ(t) = 0.

This defines a path [A(t), ψ(t)] in M_{V} for 0 < t ≤ 1. The endpoint satisfies

[A(1), ψ(1)] = [A′_{+∞}, ψ′_{+∞}],

Figure 2: The domains Ω(r), Ω_{1}(r), and Ω_{2}

where (A′_{+∞}, ψ′_{+∞}) is the limit on V as r → ∞ of the asymptotic values (A_{r}(+∞), ψ_{r}(+∞)) of (A_{r}, Ψ_{r}). Thus, we have

∂_{∞}(A′_{+∞}, ψ′_{+∞}) = a^{+},

under the boundary value map. Moreover, we obtain an element

lim_{t→0_{+}}[A(t), ψ(t)] = [A, ψ]

in M_{V}, satisfying

∂_{∞}(A, ψ) = a′_{∞},

where a′_{∞} is the asymptotic value of the limit on compact sets of the solutions (A_{r}, Ψ_{r}) on V × R, as analysed in the previous section. Similarly, we obtain a path [A(t), ψ(t)] in M_{V} for −1 ≤ t < 0, where the endpoint satisfies

[A(−1), ψ(−1)] = [A′_{−∞}, ψ′_{−∞}],

where (A′_{−∞}, ψ′_{−∞}) is the limit on V as r → ∞ of the asymptotic values (A_{r}(−∞), ψ_{r}(−∞)) of (A_{r}, Ψ_{r}). Thus, we have

∂_{∞}(A′_{−∞}, ψ′_{−∞}) = a^{−},

under the boundary value map. Similarly, the element

lim_{t→0_{−}}[A(t), ψ(t)] = [Ã, ψ̃]

in M_{V} satisfies

∂_{∞}(Ã, ψ̃) = a′_{∞}.

Notice that in general the element [A, ψ] and the element [Ã, ψ̃] need not be the same element in M_{V}. In fact, the domain Ω_{1}(r) is defined by rescaling the initial domain Ω̃(r), and the values of solutions near the frontier t = ±T_{r} of Ω̃(r) need not approach the same limits as r → ∞.

From the information on the convergence on compact sets obtained in the previous section, together with the choice of the domains, we can only derive the relation

∂_{∞}(A, ψ) = ∂_{∞}(Ã, ψ̃).

Thus, both [A, ψ] and [Ã, ψ̃] are in the zero-dimensional fiber ∂_{∞}^{−1}(a′_{∞}), cf. [4].

Moreover, by the previous analysis, together with the analysis of asymptotics on V × R of the previous section, we obtain convergence on the Ω_{2}(r) to a function a : D^{+} → χ_{0}(T^{2}, V), holomorphic in a neighborhood U of the half disk

D^{+} = {ρ ∈ (−∞, 0] θ ∈ [−π/2, π/2]},

which maps the center of the disk to a′_{∞}, and such that the restriction to the boundary component {θ ∈ {−π/2, π/2}} ⊂ ∂D^{+} gives a curve in ∂_{∞}(M_{V}) ⊂ χ_{0}(T^{2}, V) connecting a^{+}, a′_{∞}, and a^{−}. The image of the component {−π/2 ≤ θ ≤ π/2, ρ = 0} of ∂D^{+} is mapped to another curve in χ_{0}(T^{2}, V), not necessarily contained in ∂_{∞}(M_{V}), which connects the points a^{+}, a′_{∞}, and a^{−}. The image of ∂D^{+} is a closed curve in χ_{0}(T^{2}, V), not necessarily smooth across the points θ = ±π/2.

3.5 The convergence theorem

We need one more preliminary Lemma, which connects the limits on compact sets, with the non-uniform limits obtained in the previous section.

Lemma 3.9 Let (A, Ψ) be a finite energy solution of the Seiberg–Witten equations (2) on the 4-manifold V ×R with the infinite end T^{2} ×[0, ∞)×R. Let a_{∞} ∈ χ_{0}(T^{2}, V) be the radial gauge limit of (A, Ψ). Let (A(t), ψ(t)) be gauge equivalent to the original (A, Ψ), in a temporal gauge in the t ∈ R direction. Then the elements (A(t), ψ(t)) converge in the s ∈ [r_{0}, ∞) direction to a_{∞}, uniformly with respect to |t| ≥ T_{0}, and we have limits

lim_{t→+∞}(A(t), ψ(t)) = (A, ψ),

and

lim_{t→−∞}(A(t), ψ(t)) = (Ã, ψ̃),

with

∂_{∞}(A, ψ) = ∂_{∞}(Ã, ψ̃) = a_{∞}.

Proof. In order to prove the convergence in the time direction, consider the asymptotics of solutions in radial gauge studied in the previous section. From the analysis of the system of ODE’s (22), we obtained rates of decay as ρ → ∞ of the form exp(−nρ) or exp(nρ − C_{nlk}e^{ρ}), or exp(−C_{nlk}e^{ρ}), for some positive constants C_{nlk} > 0.

In the last case, at a point (s, t) along the end [r_{0}, ∞) × R, the L^{2}-norm on T^{2} is bounded by exp(−C(s^{2} + t^{2})^{1/2}). In particular, we have exponential decay in the s ∈ [r_{0}, ∞) direction, uniformly in |t| ≥ T_{0}, bounded by exp(−CT_{0}s). Thus, if we consider a temporal gauge representative (A′(t), ψ′(t)) of (A′, Ψ′), for every sequence |t_{k}| → ∞ we obtain convergence up to gauge of a subsequence to an element in M_{V}, exponentially decaying in the s ∈ [r_{0}, ∞) direction to the asymptotic value a′_{∞}, at a rate at least equal to exp(−|λ|T_{0}s).

The case of the slow decaying solutions with asymptotics e^{−nρ} gives polynomial decay in the s ∈ [r_{0}, ∞) direction for every fixed t ∈ R, at a rate

\frac{1}{(1 + (\frac{t}{s})^{2})^{n/2}} · \frac{1}{s^{n}} ≤ \frac{1}{s^{n}} ,

uniform in |t| ≥ T_{0}. Thus, in this case, the temporal gauge representative (A′(t), ψ′(t)) converges to solutions of (1) on V, as t → ±∞, that decay to the asymptotic value a′_{∞} at a polynomial rate 1/s^{n}. The analysis of [4] on the center manifold for the equations (1) on T^{2} ×[r_{0}, ∞) (cf. [15]), imply the following. If the decay is asymptotic to 1/s^{n}, with n ≥ 2, then the actual rate of decay of the limit solution is exponential and we have

a′_{∞} ∈ ∂_{∞}(M_{V}^{∗}).

If the rate of decay is 1/s, then the limit solution might also decay like 1/s along s ∈ [r_{0}, ∞). In this case the asymptotic value is a′_{∞} = ϑ, the unique “bad point” in the character variety χ(T^{2}). ⋄

We can summarize all the previous results as follows Theorem 3.10 On the manifold Y (r) × R consider classes of finite energy solutions [A_{r}, Ψ_{r}] of (2), with asymptotic values

[A_{r}(±∞), ψ_{r}(±∞)] = [A′_{±∞}, ψ′_{±∞}]#_{a^{±}}^{r}[a^{±}, 0],

as t → ±∞. Under the usual assumptions on the scalar curvature on ν(K), we obtain the following geometric limits as r → ∞.

(i) a finite energy [A′, Ψ′] on V × R, with radial limit a′_{∞} ∈ χ_{0}(T^{2}, V), and limits as t → ±∞

[A, ψ] and [Ã, ψ̃] in ∂_{∞}^{−1}(a′_{∞}) ⊂ M_{V} .

(ii) a solution [a′′_{∞}, 0] on ν(K) × R, with a′′_{∞} ∈ χ_{0}(T^{2}, ν(K)) .

(iii) a path [A(t), ψ(t)] ∈ M_{V} for t ∈ [−1, 0) ∪ (0, 1] with

[A(±1), ψ(±1)] = [A′_{±∞}, ψ′_{±∞}]

and the limits as t → 0_{±} equal to [A, ψ] and [Ã, ψ̃].

(iv) a path [A(t), 0] in M_{ν(K)} = χ(ν(K)) for t ∈ [−1, 1] satisfying [A(±1), 0] = [a^{±}, 0] and [A(0), 0] = [a′′_{∞}, 0].

(v) a holomorphic map a : D → χ_{0}(T^{2}, Y), with a(0) = a_{∞}, the limit on compact sets on the cylinder T^{2} × [−r, r] × R.

(vi) maps a_{V} : D^{+} → χ_{0}(T^{2}, V) and a_{ν} : D^{+} → χ_{0}(T^{2}, ν(K)), holomorphic on a neighborhood of the half disk D^{+} with a_{V} (ρ, ±π/2) = [A(t), ψ(t)] and a_{ν}(ρ, ±π/2) = [A(t), 0].

Proof. Suppose given a family (A_{r}, Ψ_{r}) of finite energy solutions of the Seiberg-Witten equations (2) on the manifolds Y (r) × R, with asymptotic values (A_{r}(±∞), ψ_{r}(±∞)) as t → ±∞ satisfying (1). Assume that, as r → ∞ the asymptotic values split as

(A_{r}(±∞), ψ_{r}(±∞)) = (A′_{±∞}, ψ′_{±∞})#_{a^{±}}^{r}λ(a^{±}, 0),

for some gauge transformation λ on ν(K), according to [4].

Then, up to gauge, there is a subsequence, which we still denote (A_{r}, Ψ_{r}), with the following behavior. On compact sets contained in the long cylinder T^{2} × [−r, r] × R it converges smoothly to a flat connection a_{∞} on T^{2} × R^{2}, constant in the R^{2} directions. On compact sets in V × R it converges smoothly to a finite energy solution (A′, Ψ′) of the Seiberg-Witten equations (2) on V × R, which is exponentially decaying in radial gauge on the end T^{2} × [0, ∞) × R to a flat connection a′_{∞}. This follows from the analysis of the asymptotics in the previous section.

The solution (A′, Ψ′) in a temporal gauge in the t ∈ R direction converges to solutions (A, ψ) and (Ã, ψ̃) of the Seiberg-Witten equations (1) on V, as t → ±∞ respectively. The elements (A, ψ) and (Ã, ψ̃) have asymptotic value along the end T^{2} × R of V

∂_{∞}(A, ψ) = ∂_{∞}(Ã, ψ̃) = a′_{∞},

under the map ∂_{∞} analyzed in [4]. This result follows from Lemma 3.9

Finally, again from the arguments presented in the previous sections, on compact sets in ν(K) × R the subsequence (A_{r}, Ψ_{r}) converges to a solution of the abelian ASD equation, that is, up to gauge, to a flat connection a′′_{∞} on T^{2}.

Up to the changes of coordinates ρ ↦ ρ − T_{r} and |t| ↦ e^{−T_{r}}|t|, for a sufficiently fast growing T_{r} → ∞, there is a subsequence of (A_{r}, Ψ_{r}) that converges smoothly on the domain Ω_{1} defined above to a path [A(t), ψ(t)] in M_{V} for t ∈ [−1, 0) ∪ (0, 1] with

[A(−1), ψ(−1)] = [A′_{−∞}, ψ′_{−∞}], with ∂_{∞}(A′_{−∞}, ψ′_{−∞}) = a^{−}

lim_{t→0_{−}}[A(t), ψ(t)] = [Ã, ψ̃], with ∂_{∞}(A, ψ) = a′_{∞}

lim_{t→0_{+}}[A(t), ψ(t)] = [A, ψ], with ∂_{∞}(A, ψ) = a′_{∞}

[A(1), ψ(1)] = [A′_{+∞}, ψ′_{+∞}] with ∂_{∞}(A′_{+∞}, ψ′_{+∞}) = a^{+}.

Under the same change of coordinates there is a subsequence that converges smoothly on the domain Ω_{2} described above to a function a_{V} : D^{+} → χ_{0}(T^{2}, Y) = C, holomorphic in a neighborhood U of the half disk D^{+}, which maps the center of the disk to a′_{∞} and such that the restriction to the boundary component θ ∈ {−π/2, π/2} of D^{+} gives a curve in χ_{0}(T^{2}, Y) connecting a^{+}, a′_{∞}, and a^{−}.

Similarly, on the manifold ν(K) × R we can consider similarly defined domains Ω_{1} and Ω_{2}, and an analogous reparametrization. We obtain convergence on Ω_{1} to a path [A(t), 0] in M_{ν(K)} = χ(ν(K)) for t ∈ [−1, 1] with

[A(−1), 0] = [a^{−}, 0],
[A(0), 0] = [a′′_{∞}, 0],
[A(1), 0] = [a^{+}, 0],

where a′′_{∞} is the limit on compact sets of the (A_{r}, Ψ_{r}) on ν(K) × R, up to gauge. On Ω_{2} we obtain convergence to a function a_{ν} : D^{+} → χ_{0}(T^{2}, Y), holomorphic in a neighborhood U of the half disk D^{+}, which maps the center of the disk to a′′_{∞}, and such that the restriction to the boundary component θ ∈ {−π/2, π/2} of D^{+} gives a curve in χ_{0}(T^{2}, Y) connecting a^{+}, a′′_{∞}, and a^{−}.

Finally, the reparametrization ρ ↦ ρ − T_{r} on the long cylinder (s, t) ∈ [−r, r] × R gives convergence on the disk D = {ρ ≤ 0} to a holomorphic map a : D → χ_{0}(T^{2}, Y), which maps the center of the disk to a_{∞}.

There are subdomains D_{1}^{+} and D_{2}^{+} inside the disk D, and conformal equivalences φ_{i} : D^{+} → D_{i}^{+}, such that we have

a|_{D_{1}^{+}} = a_{V} ∘ φ_{1}^{−1}
a|_{D_{2}^{+}} = a_{ν} ∘ φ_{2}^{−1}.

In the special case where a^{+} = a^{−} = a_{∞} up to gauge, the holomorphic functions are constant and the paths [A(t), ψ(t)] in M_{V} and M_{ν(K)} are also constant. ⋄

The limits described in Theorem 3.10 are illustrated in the Figure 3. Notice that, in general, the points a′_{∞}, a′′_{∞}, and a_{∞} are distinct points in

χ_{0}(T^{2}, Y). The point a′_{∞} is constrained to be on the 1-dimensional subspace ∂_{∞}(M_{V}) and the point a′′_{∞} is contained in M_{ν(K),μ}^{red} = χ(ν(K)).

We have seen that, if the three points a^{+}, a^{−}, and a_{∞} coincide, a^{+} = a^{−} = a_{∞} up to gauge, then the flow line (A_{r}, Ψ_{r}) on Y (r) × R must connect critical points which break through the same asymptotic values, a^{+} = a^{−} = a_{∞}.

We can also observe that the following Corollary holds.

Corollary 3.11 Assume that the points a′_{∞} and a′′_{∞} coincide up to gauge. Assume, moreover, that they are distinct from the bad point ϑ, and from both a^{+} and a^{−},

a^{+} ≠ a′_{∞} ≠ a^{−}.

then the original flow line (A_{r}, Ψ_{r}) on Y (r)×R must connect critical points with relative index at least two.

Proof. Let (A′, Ψ′) be the limit on compact sets in V × R of the family (A_{r}, Ψ_{r}) as r → ∞. Then the point a′_{∞} is the radial limit of (A′, Ψ′). By

Figure 3: The maps a_{ν}, a_{V}, and a in χ(T^{2})

Lemma 3.9 we know that as t → ±∞ the solution (A′, Ψ′) has asymptotic values [A, ψ] and [Ã, ψ̃] in M_{V}^{∗} with

∂_{∞}[A, ψ] = ∂_{∞}[Ã, ψ̃] = a_{∞}.

By gluing this solution with the solution (a′′_{∞}, 0) on the ν(K) × R side, we obtain a flow line in Y (r) × R which connects the critical points

[A, ψ]#_{a_{∞}}^{r}[a_{∞}, 0]

and

[Ã, ψ̃]#_{a_{∞}}^{r}[a_{∞}, 0].

This shows that there are two intermediate critical points between the endpoints of the flow line (A_{r}, Ψ_{r}). ⋄

4 Gluing

In this section we describe how to produce an approximate solution to the 4-dimensional Seiberg–Witten equations on Y (r(T)) × R, with gluing parameter T ≥ T_{0}, by pasting together the various geometric limits of Theorem 3.10.

4.1 The approximate solutions

We define the following quantities that will be used throughout this section:

R(T) = \frac{1}{π} (e^{T} + T),

ℓ(T) = \frac{1}{π} (e^{T} + T) sin (\frac{πT}{e^{T} + T} ) ,

r(T) = (r_{0} + \frac{1}{π} (e^{T} + T) cos (\frac{πT}{e^{T} + T} )) .

Moreover, we define

Υ(t, T) = r(T) − (R(T)^{2} − t^{2})^{1/2},

for |t| ≤ R(T), and

Υ̃(t, T) = r_{0} + cos (\frac{πT}{e^{T} + T} ) − min \{(1 − t^{2})^{1/2}, cos (\frac{πT}{e^{T} + T} )\} ,

for |t| ≤ 1.

Consider the path [A(t), ψ(t)] on the domain

V_{r(T)} × ([−1, −ℓ(T)/R(T)] ∪ [ℓ(T)/R(T), 1],  (35)

with T large enough, so that the elements [A(±ℓ(T)/R(T)), ψ(±ℓ(T)/R(T))] are very close to the elements [A, ψ] and [Ã, ψ̃], and with r_{0} large enough, so that at s = r_{0} the elements [A(t), ψ(t)] are very close to the asymptotic values a_{∞}(t), up to an error of the order of e^{−δr_{0}}. Consider the rescaled domain

V_{r(T)} × ([−R(T), −ℓ(T)] ∪ [ℓ(T), R(T)]),  (36)

under the change of coordinates |t| → R(T)|t|, and write [A(t), ψ(t)]_{T} for the corresponding rescaled element. Consider then the path [A(t), ψ(t)]_{T} restricted over the domain

\begin{matrix}R_{1}(T) = V ∪_{T^{2}×\{s=0\}×[−R(T),R(T)]} \\ T^{2} × \{(s, t) | s ∈ [0, Υ(t, T)], t ∈ [−R(T), R(T)]\} \end{matrix}  (37)

inside the rescaled domain (36).

By Theorem 3.10, we can consider a disk D_{T} of radius R(T), and a map a_{T} : D_{T} → χ_{0}(T^{2}, Y), such that the center of the circle is mapped to the limit a_{∞} of Theorem 3.10, and the points along the boundary corresponding to the angles {−π, −π/2, 0, π/2} are mapped to the points {a′′_{∞}, a^{−}, a′_{∞}, a^{+}}, respectively. The function a_{T} : D_{T} → χ_{0}(T^{2}, Y) agrees with the map a : D → χ_{0}(T^{2}, Y) of Theorem 3.10 restricted to a subdomain of the unit disk D homeomorphic to D_{T}. We can guarantee, by choosing T large enough, that on the arc of length 2T on ∂D_{T}, centered at the angle θ = 0, the values of the function a_{T} are sufficiently close to a′_{∞}, and on the similar arc centered at the angle θ = −π the values of the function a_{T} are sufficiently close to a′′_{∞}.

Consider then the element (A′, Ψ′) restricted over the domain

R_{3}(T) = V_{r_{0}} × [−ℓ(T), ℓ(T)],  (38)

For T ≥ T_{0} large enough, we have that, near t = ±ℓ(T), (A′, Ψ′) is sufficiently close to the elements [A, ψ] and [Ã, ψ̃], and, near s = r_{0}, we have that (A′, Ψ′) is sufficiently close to the asymptotic value a_{∞}.

Moreover, consider the rescaled path (a′′(t), 0)_{T} on the domain

\begin{matrix}R_{4}(T) = ν(K)∪_{T^{2}×\{s=0\}×[−R(T),−ℓ(T)]∪[ℓ(T),R(T)]} \\ T^{2} × \{(s, t) | s ∈ [0, Υ(t, T)], t ∈ [−R(T), −ℓ(T)] ∪ [ℓ(T), R(T)]\} \end{matrix} . (39)

Consider then the element (a′′_{∞}, 0) on the domain

R_{5}(T) = ν(K)_{r_{0}} × [−ℓ(T), ℓ(T)],  (40)

Now consider the manifold Y (r(T)) × R, with the long cylinder T^{2} × [−r(T), r(T)] × R. Consider the product region

T^{2} × {s ∈ [−r(T), r(T)], t ∈ [−R(T), R(T)]}  (41)

in Y (r(T)) × R as illustrated in part (a) of Figure 4.

Consider inside (41) the region R_{2}(T) given by T^{2} × D_{T}, with the disk D_{T} centered at s = 0, t = 0, as in (c) of Figure 4. We identify the regions R_{i}(T) of (37), (38), (39), (40), with the corresponding regions (e), (f), (b), (d), of Figure 4. The smaller strip of regions R_{3}(T) and R_{5}(T) has height

2ℓ(T) = \frac{2}{π} (e^{T} + T) sin (\frac{πT}{e^{T} + T} ) ,

and the larger horizontal strip of the regions R_{1}(T) and R_{4}(T) has height 2R(T) = \frac{2}{π} (e^{T} + T). The arc of ∂D_{T} cut out by the smaller strip has length 2T.

We construct an approximate solution on Y (r(T)) × R supported on the strip |t| ≤ R(T).

We consider smooth functions η_{T,1}(s, t), η_{T,2}(s, t), η_{T,3}(s, t), η_{T,4}(s, t), and η_{T,5}(s, t) with values in [0, 1], supported in the shaded regions in (e) (c), (f), (b) and (d) of Figure 4, respectively, such that

η_{T,1} + η_{T,2} + η_{T,3} + η_{T,4} + η_{T,5} ≡ 1

is satisfied everywhere in (41). We extend by 1 the functions

eta_{T,1}(s, t), η_{T,3}(s, t), η_{T,4}(s, t), and η_{T,5}(s, t)

on the sides V and ν(K) of the corresponding domains R_{i}(T) which are not represented in the figure.

We finally choose a smooth cutoff function χ with values in [0, 1], identically equal to 1 on the complement of |t| ∈ [R(T) − ϵ, R(T)] in (41) and equal to zero at t = ±R(T), and we define the approximate solution as

\begin{aligned}(A, Ψ)_{T} = & χ · (η_{T,1}(A(t), ψ(t))_{T} + η_{T,2}a_{T} \\ & +η_{T,3}(A′, Ψ′) + η_{T,4}(a′′(t), 0)_{T} + η_{T,5}(a′′_{∞}, 0)). \end{aligned}  (42)

4.2 Linearizations

Let L_{A(t),ψ(t)} be the linearization of the 3-dim Seiberg-Witten equations (1) at the solutions (A(t), ψ(t)). Consider the operator L_{A(t),ψ(t)} acting on pairs (α, ϕ) of L_{1}^{2} 1-forms and spinors.

First, let us recall from [4] that, given an element [A, ψ] in M_{V}^{∗}, the operator L_{A,ψ} is a Fredholm operator on the weighted Sobolev spaces of 1-forms and spinors on V,

L_{A,ψ} : L_{1,δ}^{2}(V) → L_{δ}^{2}(V),  (43)

Figure 4: The construction of the approximate solution

with

δ = \frac{1}{2} min\{|λ|, λ ∈ spec(Q_{a′_{∞}})\},

and a′_{∞} = ∂_{∞}([A, ψ]). By Theorem 6.2 and 7.4 of [10], we can consider the Fredholm operator L_{A,ψ} acting on L_{1}^{2} 1-forms and spinors. For a generic choice of the perturbation of the equations on V, as in [4], the operator

L_{A,ψ} : L_{1}^{2}(V) → L^{2}(V)  (44)

is injective and surjective. In the following section we shall discuss the effect of adding the perturbation. For the purpose of this section, let us just assume that injectivity and surjectivity are achieved. On the manifold V_{r}, with r ≥ r_{0}, we can then consider the operator L_{A,ψ} acting between

L_{A,ψ} : L_{1}^{2}(V_{r}, P_{+} ⊕ ℓ) → L^{2}(V_{r}),  (45)

that is, on the space of 1-forms and spinors on V_{r}, with boundary conditions defined by the APS condition [1]

P_{+} = L^{2}-closure of {ϕ_{k}|μ_{k} > 0},

with ϕ_{k} eigenfunctions of the operator Q_{a_{∞}} on T^{2} with eigenvalue μ_{k}, and by the Lagrangian subspace ℓ in H^{1}(T^{2}, iR) defined by the asymptotic values of the extended L^{2}-solutions of L_{A,ψ}(α, ϕ) = 0 on V. By Proposition 2.4 of [3], we have that (45) is a self-adjoint Fredholm operator with

Ker(L_{A,ψ} : L_{1}^{2}(V) → L^{2}(V)) ≅ Ker(L_{A,ψ} : L_{1}^{2}(V_{r}, P_{+} ⊕ ℓ) → L^{2}(V_{r})).

Lemma 4.1 Consider the operators L_{A(t),ψ(t)}, L_{A,ψ}, and L_{Ã,ψ̃} acting on L_{1}^{2} 1-forms and spinors on the manifold V with the infinite cylindrical end T^{2} × [0, ∞). Assume that these operators L_{A(t),ψ(t)} are surjective for all t ∈ [−1, 0) ∪ (0, 1], and that the operators L_{A,ψ} and L_{Ã,ψ̃} are also surjective. Moreover, assume also that the path [A(t), ψ(t)] and the elements [A, ψ] and [Ã, ψ̃] in M_{V} are contained in the complement of ∂_{∞}^{−1}(U_{ϑ}), where ∂_{∞} is the asymptotic value map, and U_{ϑ} is a small neighborhood of the bad point ϑ in the character variety of T^{2}. Then, for all T > T_{0}, the operator

T_{T} = R(T)^{−1} \frac{∂}{∂t} + L_{A(t),ψ(t)}

acting on pairs (α, ϕ) in the L_{1}^{2} completion of compactly supported 1-forms and spinors in the domain

\begin{matrix}S_{1}(T) = V ∪_{T^{2}×\{s=0\}×[−1,1]} \\ T^{2} × \{(s, t) | s ∈ [0, Υ̃(t, T)], t ∈ [−1, 1]\} \end{matrix}  (46)

is also surjective.

Proof. From the previous discussion we can conclude that, assuming the operators L_{A(t),ψ(t)}, L_{A,ψ} and L_{Ã,ψ̃}, with domain as in (44) are surjective, we obtain that the operator L_{A(t),ψ(t)}, with domain as in (45), with r = Υ̃(t, T), has trivial kernel and is surjective. Similarly, the operators L_{A,ψ} and L_{Ã,ψ̃}, with domain as in (45), with r = r_{0}, have trivial kernel and are surjective.

For simplicity of notation, we just write L for the surjective operators we are considering. Given any h, we want to find a function f in the domain satisfying

h = (\frac{1}{R(T)} \frac{∂}{∂t} + L)f.

We know that we have h = Lg for some g. Moreover, we have an estimate

∥g(t)∥_{L^{2}(V_{Υ̃(t,T)}×{t})} ≤ C_{t}∥h(t)∥_{L^{2}(V_{Υ̃(t,T)}×{t})},

which follows from Parseval’s formula (cf. [1] pg.50), where the constant C_{t} is determined by the smallest absolute value of the non-zero eigenvalues of the asymptotic operator Q_{a_{∞}(t)},

C_{t} = \frac{c}{λ_{∞}(t)},

with

λ_{∞}(t) = min{|λ| | λ ∈ spec(Q_{a_{∞}(t)})}.

In the character variety χ(T^{2}) consider a small open neighborhood U_{ϑ} of the “bad point” ϑ (see [4]). We have the asymptotic value map

∂_{∞} : M_{V} → χ_{0}(T^{2}, V)

defined as in [4], with

∂_{∞}(A(t), ψ(t)) = a_{∞}(t).

Notice that, if we assume that the path (A(t), ψ(t)), for 0 < t ≤ 1 is in the complement of ∂_{∞}^{−1}(U_{ϑ}), then the constant C_{t} can be replaced with the uniform constant

C = c · ( min_{0≤t≤1} λ_{∞}(t))^{−1},

independent of t and of the gluing parameter T.

So we can write

h = (\frac{1}{R(T)} \frac{∂}{∂t} + L)g − \frac{1}{R(T)} \frac{∂}{∂t}g.

Again by the surjectivity of L we can write

−\frac{∂}{∂t}g = Lg_{1}.

Again, we have an estimate ∥g_{1}∥ ≤ C∥g∥, hence iterating the process we get

h = (\frac{1}{R(T)} \frac{∂}{∂t} + L) (∑_{k=0}^{n}(−1)^{k} (\frac{1}{R(T)})^{k} g_{k}) + (−1)^{n+1} (\frac{1}{R(T)})^{n+1} \frac{∂}{∂t}g_{n}.

For all T ≥ T_{0} we obtain

∑_{k=0}^{∞} (\frac{1}{R(T)})^{k} C^{k} < ∞,

since R(T) ∼ e^{T} for large T ≥ T_{0}. Thus, as n → ∞, we have uniform convergence of the series

∑_{k=0}^{n}(−1)^{k} (\frac{1}{R(T)})^{k} g_{k} → f,

and of the sequence

(−1)^{n+1} (\frac{1}{R(T)})^{n+1} \frac{∂}{∂t}g_{n} → 0.

⋄ Over the unit disk D, consider the operator

D_{1} = ( \begin{matrix}∂_{ρ} − ∗∂_{θ} & ∗d \\ −e^{2ρ} ∗ d & ∂_{ρ} \end{matrix} )

acting on pairs (ã, f̃) where ã(w, ρ, θ) is 1-form and f̃(w, ρ, θ) is a function. Consider also the operator

D_{2} = ( \begin{matrix}\frac{1}{R(T)}(∂_{ρ} − i∂_{θ}) & −e^{ρ+iθ}i¯∂_{a}^{∗} \\ e^{ρ−iθ}i¯∂_{a} & \frac{1}{R(T)}(∂_{ρ} + i∂_{θ}) \end{matrix} ) ,

acting on a spinor (α̃, β̃). Here a(ρ, θ) is the image under the map a : D → χ_{0}(T^{2}, Y) of Theorem 3.10, restricted to a subdomain of D homeomorphic to the disk. That is, for each (ρ, θ), a(ρ, θ) is a flat connection on T^{2}.

Lemma 4.2 Without loss of generality, we can assume that the image of the map a : D → χ_{0}(T^{2}, Y) avoids the lattice of bad points ϑ in χ_{0}(T^{2}, Y)

Proof. We have unique bad point ϑ in the character variety of T^{2}, [4], which corresponds to a lattice of bad points in the universal cover χ_{0}(T^{2}, Y).

First notice that, according to the proof of Lemma 3.9, the condition that the endpoint a′_{∞} is the bad point ϑ is non-generic: by direct inspection of the characteristic polynomial of the system (29) this case corresponds to the vanishing of the coefficients C_{nlk} that give the faster decay e^{±nρ−C_{nlk}e^{ρ}}. Thus, under generic choice of the perturbation, as will be discussed in the next section, we can assume that a′_{∞} ≠ ϑ.

This implies that, under the maps a_{V} and a_{ν} of Theorem 3.10, the image of the side θ ∈ {−π/2, π/2} of the boundary of the half disk D^{+} is a smooth path in ∂_{∞}M_{V} or in ∂_{∞}M_{ν(K)}, connecting the points a^{±} which avoids the lattice of bad points. We can always replace the domain D^{+} with some smaller domain containing the same component θ ∈ {−π/2, π/2} and ρ ∈ (−∞, 0] of the boundary, in such a way that the image of the rest of the domain also avoids the lattice of bad points. That still determines uniquely the holomorphic function. Thus, we can assume for simplicity that the image of the subdomain of D under the map a : D → χ_{0}(T^{2}, Y) also contains no bad point ϑ in χ_{0}(T^{2}, Y). ⋄

We want to consider D_{1}⊕D_{2} acting on elements (ã, f̃, α̃, β̃) that represent deformations of (a, 0, 0, 0) in the L_{1}^{2} completion of compactly supported 1forms and spinors on D.

Lemma 4.3 The operator D_{1}⊕D_{2} acting on the L_{1}^{2} completion of compactly supported 1-forms and spinors on D, has trivial Cokernel.

Proof. We know [4] that the operator

( \begin{matrix}0 & −e^{ρ+iθ} ¯∂_{a}^{∗} \\ e^{ρ−iθ} ¯∂_{a} & 0 \end{matrix} )

is surjective if the flat connection a(ρ, θ) avoids the lattice of bad points ϑ in χ_{0}(T^{2}, Y). Then, by the same argument of Lemma 4.1, we obtain that for sufficiently large T ≥ T_{0}, the operator D_{2} is surjective.

Suppose given a pair (β, h) of a 1-form and a function, such that (β, h) is orthogonal to the range of D_{1}. Then, the pair (β, h) satisfies the equations

∂_{ρ}h = (R(T))^{−2}d^{∗}β
∂_{ρ}β = ∗∂_{θ}β + ∗d(e^{−2ρ}h).

The Fourier coefficients satisfy the ODE

\begin{matrix}h′_{nlk} = (R(T))^{−2}(−ilv_{nlk} + inu_{nlk}) \\ u′_{nlk} = inv_{nlk} − e^{−2ρ}ikh_{nlk} \\ v′_{nlk} = −inu_{nlk} + e^{−2ρ}ilh_{nlk}. \end{matrix}  (47)

We can show that solutions of (47) diverge sufficiently fast at ρ → −∞, so that they fail to be in L^{2}. Since we are interested in the ρ → −∞ behavior, we can isolate in the system (47) a leading term given by the system

\begin{matrix}h′_{nlk} = (R(T))^{−2}(−ilv_{nlk} + inu_{nlk}) \\ u′_{nlk} = −e^{−2ρ}ikh_{nlk} \\ v′_{nlk} = e^{−2ρ}ilh_{nlk}, \end{matrix}  (48)

and treat the remaining terms as a perturbation.

Solutions of the system (48) are solutions of

h′′_{nlk} = e^{−2ρ}(R(T))^{−2}(k^{2} + l^{2})h_{nlk}.

These are combinations of Bessel functions of type I(0, z) and K(0, z), with the variable z = c_{nlk}\sqrt{(R(T))^{−2}e^{−2ρ}}, for some constants c_{nlk} > 0. None of these functions is in L^{2}(D). ⋄

Consider the operator T_{a′′(t)} of the form

\frac{1}{R(T)} \frac{∂}{∂t} + ( \begin{matrix}∗d & −d & 0 \\ −d^{∗} & 0 & 0 \\ 0 & 0 & ∂_{a′′(t)} \end{matrix} ) ,  (49)

where ∂_{a′′(t)} is the 3-dimensional Dirac operator, twisted with the flat connection

a′′(w, s, t) = a′′(t)(w, s).

Lemma 4.4 If the path a′′(t) of connections in χ_{0}(T^{2}, Y) avoids the lattice of bad points, then the operator T_{a′′(t)}, acting on the L_{1}^{2} completion of the space of compactly supported 1-forms and spinors on the domain

\begin{matrix}S_{4}(T) = ν(K)∪_{T^{2}×\{s=0\}×[−1,1]} \\ T^{2} × \{(s, t) | s ∈ [0, Υ̃(t, T)], t ∈ [−1, 1]\} \end{matrix}  (50)

is surjective for sufficiently large T ≥ T_{0}.

Proof. The operator

\frac{1}{R(T)} \frac{∂}{∂t} + ∂_{a′′(t)}

is surjective, for sufficiently large T ≥ T_{0}, by the argument of Lemma 4.1. Suppose given an element (β, h) orthogonal to the range of

\frac{1}{R(T)} \frac{∂}{∂t} + ( \begin{matrix}∗d & −d \\ −d^{∗} & 0 \end{matrix} ) .

In the large T >> 0 limit, the element (β, h) satisfies

d^{∗}β = 0 and ∗ dβ − dh = 0.

This implies that h is harmonic, in the L_{1}^{2} completion of the space of compactly supported function, hence it is vanishing by the maximum principle. On the other hand β satisfies (d + d^{∗})β = 0, which also implies β ≡ 0 in the L_{1}^{2} completion of the space of compactly supported forms on (50). ⋄

4.3 The solutions

In this section we prove that every approximate solution can be deformed to an actual solution of the Seiberg–Witten equations. We first prove that the linearization D_{T} at the approximate solution is surjective. Here the operator

D_{T} = D_{A,Ψ}

is the linearization of the 4-dimensional Seiberg–Witten equations (2) on Y (r(T)) × R, and (A, Ψ) is the approximate solution (42). Then, the gluing theorem follows as a fixed point argument, cf. similar arguments in [4], [5], [7], [11], [17].

Our case here is similar to the construction of [5], since we want to prove a gluing theorem where the underlying geometry also changes along with the gluing parameter T. In particular, we will obtain the actual solution close to the approximate solution by a fixed point arguments with constants depending on T, as in [5].

In particular, we have seen from the convergence result of Theorem 3.10, that the non-uniform geometric limits are obtained from solutions on Y (r)× R after suitable rescaling of the coordinates. Thus, in order to prove the gluing theorem, we will introduce a norm on the domain Y (r(T))×R which takes into account the necessary rescaling. That will allow us to compare the operator D_{T} with the operators:

D_{1} = R(T)^{−1} \frac{∂}{∂t} + L_{A(t),ψ(t)}

on the domain S_{1}(T) of (46),

D_{2} = D_{1} ⊕ D_{2}

on the unit disk S_{2}(T) = D,

D_{3} = D_{A′,Ψ′},

the linearization of the equations (2) at the solution (A′, Ψ′) on the domain S_{3}(T) = R_{3}(T),

D_{4} = R(T)^{−1} \frac{∂}{∂t} + L_{(a′′(t),0)}

as in (49), on the domain S_{4}(T) defined in (50), and

D_{5} = D_{(a′′_{∞},0)},

on the domain S_{5}(T) = R_{5}(T).

With the results of the Lemmata 4.1, 4.3, and 4.4, and with the perturbation that we shall discuss in the next section, we will be able to guarantee that the operators D_{i} have trivial Cokernel. We take this as an assumption in the remaining of this section.

We can now introduce a T -dependent Banach norm, which takes into account the effect of rescaling. We introduce the notation

ϵ(T) = \frac{1}{R(T)}

θ(T) = \frac{πT}{e^{T} + T} .

We write connections and spinors as

a(w, s, t) + f(w, s, t)dt + h(w, s, t)ds

and

(α(w, s, t), β(w, s, t))

in the regions R_{1}(T), R_{3}(T), R_{4}(T), and R_{5}(T). On R_{2}(T) we write them as

a(w, ρ, θ) = a(w, e^{ρ+iθ})
f(w, ρ, θ) = e^{−ρ} cos θ h(w, e^{ρ+iθ}) − e^{−ρ} sin θ f(w, e^{ρ+iθ})  (51)

h(w, ρ, θ) = e^{−ρ} cos θ f(w, e^{ρ+iθ}) + e^{−ρ} sin θ h(w, e^{ρ+iθ})  (52)
α(w, ρ, θ) = α(w, e^{ρ+iθ})
β(w, ρ, θ) = β(w, e^{ρ+iθ})

according to the change of coordinates (20).

We define the T -rescaled L^{2}-norm as

\begin{matrix}∥(η_{T,3} + η_{T,5}) (a, f, h, α, β)∥_{L^{2}(\{(w,s,t)∈R_{3}(T)∪R_{5}(T)\})}^{2}+ \\ ∥(η_{T,1} + η_{T,4}) (a, R(T)^{1/2}f, h, α, β)∥_{L^{2}(\{(w,s,t)∈R_{1}(T)∪R_{4}(T)\})}^{2}+ \\ ∥η_{T,2} (a, R(T)^{1/2}f, R(T)^{1/2}h, α, β)∥_{L^{2}(\{(w,ρ,θ)∈R_{2}(T)\})}^{2} \end{matrix}  (53)

with 1/ϵ(T) = R(T). We denote this norm by ∥ · ∥_{0,T}.

Notice that we have to use a different rescaling on the regions R_{1}(T) ∪ R_{4}(T) and R_{2}(T). This corresponds to the rescaling used in the section on the geometric limits. In the regions R_{1}(T) ∪ R_{4}(T) we are only rescaling the temporal direction, hence in the T -dependent norm we have a weight only on the f component, whereas in the region R_{2}(T), the translation of the radial coordinates corresponds to a rescaling of both the f and the h component, due to the relations (51) and (52).

Similarly, we define the T -rescaled L_{1}^{2}-norm ∥ · ∥_{1,T} as

\begin{matrix}∥(η_{T,3} + η_{T,5}) (a, f, h, α, β)∥_{L_{1}^{2}(\{(w,s,t)∈R_{3}(T)∪R_{5}(T)\})}^{2}+ \\ R(T)∥(η_{T,3} + η_{T,5}) ∂_{t}(a, f, h, α, β)∥_{L^{2}(\{(w,s,t)∈R_{3}(T)∪R_{5}(T)\})}^{2}+ \\ ∥(η_{T,3} + η_{T,5}) ∂_{s}(a, f, h, α, β)∥_{L^{2}(\{(w,s,t)∈R_{3}(T)∪R_{5}(T)\})}^{2}+ \\ ∥(η_{T,1} + η_{T,4}) (a, R(T)^{1/2}f, h, α, β)∥_{L_{1}^{2}(\{(w,s,t)∈R_{1}(T)∪R_{4}(T)\})}^{2}+ \\ R(T)∥(η_{T,1} + η_{T,4}) d_{t}(a, R(T)^{1/2}f, h, α, β)∥_{L^{2}(\{(w,s,t)∈R_{1}(T)∪R_{4}(T)\})}^{2}+ \\ ∥(η_{T,1} + η_{T,4}) d_{s}(a, R(T)^{1/2}f, h, α, β)∥_{L^{2}(\{(w,s,t)∈R_{1}(T)∪R_{4}(T)\})}^{2}+ \\ ∥η_{T,2} (a, R(T)^{1/2}f, R(T)^{1/2}h, α, β)∥_{L_{1}^{2}(\{(w,ρ,θ)∈R_{2}(T)\})}^{2}+ \\ ∥η_{T,2} d_{ρ}(a, R(T)^{1/2}f, R(T)^{1/2}h, α, β)∥_{L^{2}(\{(w,ρ,θ)∈R_{2}(T)\})}^{2}+ \\ R(T)∥η_{T,2} d_{θ}(a, R(T)^{1/2}f, R(T)^{1/2}h, α, β)∥_{L^{2}(\{(w,ρ,θ)∈R_{2}(T)\})}^{2} \end{matrix} (54)

We can now prove the main Lemma for the gluing theorem: this can be regarded as an analogue, in our context, of Lemma 4.4 and 4.5 of [5].

Lemma 4.5 We assume that the operators D_{i} defined on the respective domains are surjective. There are constants T_{0} and C > 0, independent of T, such that we have

∥ξ∥_{1,T} ≤ C∥D_{T}^{∗} ξ∥_{0,T} ,

where D_{T}^{∗} is the adjoint with respect to the T -dependent norm. Moreover, we have

∥ξ∥_{1,T} ≤ C∥D_{T} ξ∥_{0,T} ,

for all ξ in

(∏_{i=1}^{5} Ker(D_{i}))^{⊥} .

Proof. We prove the second statement in the Lemma. The proof of the first statement is analogous, under the assumption of surjectivity of the operators D_{i}.

We first notice the following fact, which follows from the discussion of the non-uniform geometric limits in the section preceding the proof of Theorem 3.10. Each domain S_{i}(T) is obtained by a rescaling (possibly trivial) of the corresponding domain R_{i}(T). We denote by D̃_{T,i} the operator obtained from D_{T}, under the change of coordinates that produces the necessary rescaling of the domain R_{i}(T). We observe that in the operator norm over each S_{i}(T) we have

∥D̃_{T,i} − D_{i}∥ → 0

as T → ∞.

The argument then is similar to [17] §2.5. Suppose given a sequence ξ_{k}, and parameters T_{k} → ∞, satisfying

∥D_{T_{k}}ξ_{k}∥_{0,T_{k}} → 0,

and for all k

∥ξ_{k}∥_{1,T_{k}} = 1.

Moreover, if we write ξ̃_{k} for the elements ξ_{k} in the rescaled coordinates on the domains S_{i}(T), we assume that the elements ξ̃_{k} are in

(∏_{i=1}^{5} Ker(D_{i}))^{⊥} .

Assume that the cutoff functions η_{T,i} satisfy the condition

sup_{R_{i}(T_{k})} |∇η_{T_{k},i}| ≤ Cϵ_{k},

for a sequence ϵ_{k} → 0 as k → ∞.

We then have the following estimates:

∥D_{i}η̃_{T_{k},i}ξ̃_{k}∥_{L^{2}(S_{i}(T))} ≤ ∥D_{T_{k}}η_{T_{k},i}ξ_{k}∥_{0,T_{k}}
≤ Cϵ_{k} + ∥η_{T_{k},i}D_{T_{k}}ξ_{k}∥_{0,T_{k}}
≤ Cϵ_{k} + ∥D̃_{T,i} − D_{i}∥ ∥ξ_{k}∥_{1,T_{k}} + ∥D_{T_{k}}ξ_{k}∥_{0,T_{k}},

where all the terms in the last expression converge to zero.

Thus, we have obtained

∥D_{i}η̃_{T_{k},i}ξ̃_{k}∥ → 0,

hence, by the assumptions on ξ_{k}, we have

η̃_{T_{k},i}ξ̃_{k} → ξ_{i},

with ξ_{i} ∈ Ker(D_{i}).

We then derive the contradiction

1 = ⟨ξ_{k}, ξ_{k}⟩_{1,T} = lim_{k→∞}⟨ξ_{k}, ξ_{k}⟩_{1,T}

= lim_{k→∞} ∑_{i=1}^{5}⟨η_{T_{k},i}ξ_{k}, ξ_{k}⟩_{1,T} = lim_{k→∞} ∑_{i=1}^{5}⟨η̃_{T_{k},i}ξ̃_{k}, ξ̃_{k}⟩ = lim_{k→∞} ∑_{i=1}^{5}⟨ξ_{i}, ξ̃_{k}⟩ = 0.

⋄

Now we can state the second main result of this part of the work, namely the gluing theorem.

Theorem 4.6 Given any approximate solution Ξ_{0} = (A, Ψ), constructed as in (42), for all sufficiently large T ≥ T_{0}, there exists a solution Ξ of the equations (2) on Y (r(T))×R, satisfying ∥Ξ−Ξ_{0}∥_{1,T} ≤ cϵ(T), for a constant c > 0 independent of T ≥ T_{0}.

Proof. We divide the proof in several steps, following the similar argument of [5].

On the domain (41), consider the map

σ_{T} (a, f, h, α, β) = \{ \begin{matrix}∂_{t}a − dh + ∗(∂_{s}a − df) − ∗i(ᾱβ + αβ̄) \\ ∂_{t}f − ∂_{s}h + ∗(F_{a} − \frac{i}{2}(|α|^{2} − |β|^{2})ω) \\ ∂_{t}α + hα + i∂_{s}α + ifα + ¯∂_{a}^{∗}β \\ ∂_{t}β + hβ − i∂_{s}β + ¯∂_{a}α. \end{matrix}

Lemma 4.7 Let Ξ_{0} be the approximate solution as in (42). The estimate

∥σ_{T} (Ξ_{0})∥_{0,T} ≤ c_{0}ϵ(T)^{2}

is satisfied, with c_{0} > 0 independent of T ≥ T_{0}.

Proof. We have σ_{T} (Ξ_{0})(w, s, t) = 0 except on the overlap between the supports of the functions η_{T,i} used in (42). The functions η_{T,i} can be chosen so that, after rescaling, these regions have total area bounded by Cϵ(T)^{2}. We then introduce the constant

C′ = max \{ \begin{matrix}∥(A, ψ)∥_{L^{∞}(V)}, ∥(Ã, ψ̃)∥_{L^{∞}(V)}, \\ sup_{0≤t≤1} ∥a′(t)∥_{L^{∞}(T^{2})}, sup_{0≤t≤1} ∥a′′(t)∥_{L^{∞}(T^{2})} \end{matrix} \} .

This yields the estimate of the Lemma, with a constant c_{0} > CC′. ⋄

Now define elements as follows

Ξ_{1} = Ξ_{0} + ξ_{0} ξ_{0} = D_{T}^{∗} η_{0} D_{T} D_{T}^{∗} η_{0} = −σ_{T} (Ξ_{0}).

The elements are well defined because of the following.

Lemma 4.8 The equation D_{T} D_{T}^{∗} η_{0} = −σ_{T} (Ξ_{0}) admits a unique solution η_{0} with ∥η_{0}∥_{1,T} ≤ C. Moreover, we have

∥ξ_{0}∥_{1,T} ≤ c_{1}∥σ_{T} (Ξ_{0})∥_{0,T} ≤ c_{0}c_{1}ϵ(T)^{2},

with c_{1} > 0 independent of T ≥ T_{0}.

Proof. The operator D_{T} D_{T}^{∗} is invertible, under the assumptions of Lemma 4.5. The estimate then follows from the previous Lemma 4.5 and Lemma 4.7.

⋄ The linearization of the map σ_{T} is given by

dσ_{T,Ξ_{0}}(ã, f̃, h̃, α̃, β̃) =

\{ \begin{matrix}∂_{t}ã − dh̃ + ∗(∂_{s}ã − df̃) − ∗i(ᾱβ̃ + \overline{α̃}β + α̃β̄ + α\overline{β̃}) \\ ∂_{t}f̃− ∂_{s}h̃ + ∗(d_{T^{2}} ã − \frac{i}{2}(2Re(ᾱα̃) − 2Re(β̄β̃))ω) \\ ∂_{t}α̃ + hα̃ + h̃α + i∂_{s}α̃ + if̃α + ifα̃ + ¯∂_{a}^{∗}β̃ + ¯∂∗ ; ãβ \\ ∂_{t}β̃ + h̃β + hβ̃ − i∂_{s}β̃ + ¯∂_{a}α̃ + ¯∂ãα. \end{matrix}

By construction we have

dσ_{T,Ξ_{0}}(ξ) = D_{T} (ξ).

Consider the non-linear part of the map σ_{T}, that is the expression

N σ_{T} (ξ) = σ_{T} (Ξ_{0} + ξ) − σ_{T} (Ξ_{0}) − dσ_{T,Ξ_{0}}(ξ).

We have

σ_{T} (Ξ_{1}) = σ_{T} (Ξ_{0} + ξ_{0}) − σ_{T} (Ξ_{0}) − dσ_{T,Ξ_{0}}(ξ_{0}) = N σ_{T} (ξ_{0}).

Lemma 4.9 We have an estimate

∥σ_{T} (Ξ_{1})∥_{0,T} ≤ c_{2}ϵ(T)^{−2}∥ξ_{0}∥_{0,T}^{2} ≤ c_{0}c_{1}c_{2}∥ξ_{0}∥_{0,T} .

Proof. The non-linear part is given by

N σ_{T} (ã, f̃, h̃, α̃, β̃) = \{ \begin{matrix}∗i(\overline{α̃}β̃ + α̃\overline{β̃}) \\ ∗\frac{i}{2}(|α̃|^{2} − |β̃|^{2})ω \\ h̃α̃ + if̃α̃ + ã.β̃ \\ h̃β̃ − if̃β̃ + ã.α̃. \end{matrix}

We have, in the L^{2}-norms

∥N σ_{T} (ξ_{0})∥ ≤ c_{2}∥ξ_{0}∥^{2} hence, in the T -dependent norms, taking into account the effect of rescaling, we obtain

∥N σ_{T} (ξ_{0})∥_{0,T} ≤ c_{2}ϵ(T)^{−2}∥ξ_{0}∥_{0,T}^{2} .

This proves the claim. ⋄

We then proceed inductively, as in [5]. We set

Ξ_{ν+1} = Ξ_{ν} + ξ_{ν} ξ_{ν} = D_{T}^{∗} η_{ν} D_{T} D_{T}^{∗} η_{ν} = −σ_{T} (Ξ_{ν}).

Lemma 4.10 The following estimates hold:

∥ξ_{ν}∥_{1,T} ≤ 2c_{1}∥σ_{T} (Ξ_{ν})∥_{0,T} ,
∥ξ_{ν}∥_{1,T} ≤ 2^{−ν}c_{0}c_{1}ϵ(T)^{2},
∥σ_{T} (Ξ_{ν+1}∥_{0,T} ≤ c_{3}∥ξ_{n}u∥_{0,T} .

Proof. The proof is completely analogous to the similar claim in [5]. Namely, the first inequality follows from Lemma 4.5; the second inequality follows from the first, together with the previous step of the induction, and the third inequality follows by deriving

∥Ξ_{ν} − Ξ_{0}∥_{1,T} ≤ ∑_{j=0}^{ν−1} ∥ξ_{j}∥_{1,T} ≤ 2c_{0}c_{1}ϵ(T)^{2}

from the second inequality, and then estimating

∥σ_{T} (Ξ_{ν+1})∥_{0,T} ≤ ∥σ_{T} (Ξ_{ν} + ξ_{ν}) − σ_{T} (Ξ_{ν}) − dσ_{T,Ξ_{0}}(ξ_{ν})∥+
∥dσ_{T,Ξ_{ν}} − dσ_{T,Ξ_{0}}∥ · ∥ξ_{ν}∥
≤ c_{4}(∥ξ_{ν}∥ + ∥Ξ_{ν} − Ξ_{0}∥)∥ξ_{ν}∥,

as in [5]. ⋄

This completes the proof of the Theorem. In fact the estimates of Lemma 4.10 guarantee that the successive approximations converge. ⋄

5 Perturbation

The problem with the gluing theorem stated in the previous section is that it relies on the assumption that the linearization L_{A(t),ψ(t)} is surjective, for all the elements [A(t), ψ(t)] in M_{V}, as in Lemma 4.1. We know that, in general, these conditions are satisfied only after introducing a suitable perturbation.

In [4] we defined a class of perturbations P of the Chern–Simons–Dirac functional, such that, for a generic element P = (U, V) ∈ P, the operators L_{A,ψ} and D_{A,Ψ} are both surjective. These operators, in the perturbed case, are the linearizations of the corresponding perturbed critical point equation

\{ \begin{matrix}∗F_{A} = σ(ψ, ψ) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} μ_{j} \\ ∂_{A}(ψ) + ∑_{j=1}^{K} \frac{∂V}{∂ζ_{j}} ν_{j}.ψ = 0. \end{matrix}

and of the perturbed flow line equation

\{ \begin{matrix}\frac{∂A}{∂t} & = & − ∗ F_{A} + σ(ψ, ψ) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} μ_{j} \\ \frac{∂ψ}{∂t} & = & −∂_{A}ψ − ∑_{j=1}^{K} \frac{∂V}{∂ζ_{j}} ν_{j}.ψ, \end{matrix}

respectively.

For simplicity, we use the notation

P_{U} = ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} μ_{j},

P_{V} = ∑_{j=1}^{K} \frac{∂V}{∂ζ_{j}} ν_{j}.

Thus, we have

\{ \begin{matrix}∗F_{A} = σ(ψ, ψ) + P_{U} \\ ∂_{A}(ψ) + P_{V} .ψ \end{matrix}  (55)

and

\{ \begin{matrix}\frac{∂A}{∂t} = − ∗ F_{A} + σ(ψ, ψ) + P_{U} \\ \frac{∂ψ}{∂t} = −∂_{A}ψ − P_{V} .ψ. \end{matrix}  (56)

On the manifold Y (r) with a long cylinder, and on the non-compact manifold V with an infinite cylinder, we refine the definition of the class of perturbations, as in [4], by including the requirement that the perturbation is exponentially small in the region T^{2} × [−r, r] ∪ ν(K), and on the end T^{2} × [0, ∞). We denote the corresponding class with P_{δ}, where δ is the rate of decay, depending on the smallest absolute value of the non-trivial eigenvalues of the asymptotic operator Q_{a_{∞}} on T^{2}, cf. [4].

We need to check that the main results in the previous sections can be extended to the case where the perturbation P ∈ P_{δ} is introduced.

5.1 Estimates in the perturbed case

In order to extend the results of Section 2 to the case of the equations (56), we need the analogue of the uniform estimate on the energy, and then of Lemma 2.3 and Lemma 2.5. The results of Section 2 then extend to this case without significant changes.

Lemma 5.1 Let (A_{r}, Ψ_{r}) be a finite energy solution of (56) on Y (r) × R. Then, for r ≥ r_{0}, and for any interval [t_{0}, t_{1}] of length ℓ = t_{1} − t_{0}, the estimates of Lemma 2.3 and Lemma 2.5 hold, with

s_{0} = max_{Y (r_{0})}{−s(x) + C(P), 0},

where C(P) is a positive constant depending only on the perturbation P ∈ P_{δ}.

Proof. Recall that the class of perturbations P_{δ} is defined [4] by considering complete L^{2} bases {ν_{i}(r)}_{i=1}^{∞} and {μ_{j}(r)}_{j=1}^{∞}, satisfying

sup_{T^{2}×[−r,r]} |ν_{i}(r)| ≤ sup_{T^{2}×[−r_{0},r_{0}]} |ν_{i}(r_{0})|

sup_{T^{2}×[−r,r]} |μ_{j}(r)| ≤ sup_{T^{2}×[−r_{0},r_{0}]} |μ_{j}(r_{0})|,

and rescaling them with a function

f_{r}(s) = e^{−δ(s+r)},

for −r + ϵ ≤ s ≤ r − ϵ, on the cylinder T^{2} × [−r, r], with the weight δ satisfying

δ ≥ \frac{1}{2}min\{λ_{a_{∞}}|a_{∞} ∈ χ(T^{2})\U_{ϑ}\}.

The elements {f_{r}ν_{i}(r)}_{i=1}^{∞} and {f_{r}μ_{j}(r)}_{j=1}^{∞} still give complete bases, which we use to define the perturbation P_{r} in the class P_{δ} on Y (r).

Thus, the pointwise estimate obtained from the Weitzenböck formula gives

\begin{matrix}0 ≥ \frac{s}{2}|ψ|^{2} − ⟨∗F_{A}.ψ, ψ⟩ + ⟨\frac{∂U}{∂ζ_{i}} f_{r}ν_{i}(r).ψ, \frac{∂U}{∂ζ_{j}} f_{r}ν_{i}(r).ψ⟩ \\ −⟨\frac{∂U}{∂ζ_{i}} d(f_{r}ν_{i}(r)).ψ, ψ⟩ ≥ \frac{s}{2}|ψ|^{2} + \frac{1}{2}|ψ|^{4} − C(P_{r})|ψ|^{2}. \end{matrix}  (57)

The constant satisfies C(P_{r}) ≤ C(P_{0}).

We can prove a uniform bound on the energy. In fact, the energy is now defined as the variation of the perturbed Chern–Simons–Dirac functional

CSD_{P} (A, ψ) = CSD(A, ψ) + U(τ_{1}(A, ψ), ⋯, τ_{N}(A, ψ))
+V (ζ_{1}(A, ψ), ⋯, ζ_{K}(A, ψ)),

as in [4].

If (A_{r}, Ψ_{r}) is a family of finite energy solutions on Y (r)×R, with asymptotic values (A_{r}(±∞), ψ_{r}(±∞)) as t → ±∞ satisfying the perturbed equations (55), the energy

E_{r} = CSD_{P} (A_{r}(−∞), ψ_{r}(−∞)) − CSD_{P} (A_{r}(+∞), ψ_{r}(+∞))

is given by

\frac{1}{2} ∫_{Y (r)} F_{A_{r}} ∧ F_{A_{r}} + ∫_{Y (r)}⟨ψ_{r}(−∞), f_{r} ∑ \frac{∂V_{r}}{∂ζ_{j}} ν_{j}(r).ψ_{r}(−∞)⟩

− ∫_{Y (r)}⟨ψ_{r}(+∞), f_{r} ∑ \frac{∂V_{r}}{∂ζ_{j}} ν_{j}(r).ψ_{r}(+∞)⟩

+U(τ_{1}(A_{r}(−∞), ψ_{r}(−∞)), ⋯, τ_{N}(A_{r}(−∞), ψ_{r}(−∞)))
−U(τ_{1}(A_{r}(+∞), ψ_{r}(+∞)), ⋯, τ_{N}(A_{r}(+∞), ψ_{r}(+∞)))
+V (ζ_{1}(A_{r}(−∞), ψ_{r}(−∞)), ⋯, ζ_{K}(A_{r}(−∞), ψ_{r}(−∞)))
−V (ζ_{1}(A_{r}(+∞), ψ_{r}(+∞)), ⋯, ζ_{K}(A_{r}(+∞), ψ_{r}(+∞))).

We know the first term is uniformly bounded, from the analysis of the unperturbed case. Using the notation

P_{V} (r) = f_{r} ∑ \frac{∂V_{r}}{∂ζ_{j}} ν_{j}(r),

the second and third term can be estimated by writing

\begin{matrix}| ∫_{Y (r)}⟨ψ_{r}, P_{V} (r).ψ_{r}⟩dv| ≤ ∫_{Y (r_{0})} |⟨ψ, P_{V} (r).ψ⟩|dv \\ + ∫_{T^{2}×([−r,−r_{0}]∪[r_{0},r])} |⟨ψ_{r}, P_{V} (r).ψ_{r}⟩|dv. \end{matrix}  (58)

The first term in the right hand side is bounded by

∥σ(ψ_{r}, ψ_{r})∥_{L^{2}(Y_{(}r_{0}))} · ∥P_{V} (r)∥_{L^{2}(Y (r_{0}))} ≤ C C(P_{0})^{2}∥P_{V} (r_{0})∥_{L^{2}(Y (r_{0}))},

where the terms on the right are obtained using the uniform pointwise estimate (57). The second term in the right hand side of (58) can be estimated similarly by

C(r − r_{0})^{2} C(P_{0})^{2}e^{−δ(r−r_{0})} sup |ν_{i}(r_{0})|,

where the factor (r − r_{0})^{2} comes from factoring out the volume of the cylinder in the estimate of both σ(ψ_{r}, ψ_{r}) and P_{V} (r). Again we have used the pointwise estimate (57).

The remaining terms in the variation of CSD_{P} can be bounded as follows. We have

|U(τ_{j}(A_{r}(−∞), ψ_{r}(−∞))) − U(τ_{j}(A_{r}(+∞), ψ_{r}(+∞)))|
≤ |τ_{j}(A_{r}(−∞), ψ_{r}(−∞)) − τ_{j}(A_{r}(+∞), ψ_{r}(+∞))|·

∥\frac{∂U(r)}{∂τ_{j}}μ_{j}(r)∥_{L^{2}(Y_{r})}.

The last term is bounded uniformly, by our assumptions on the perturbation. We have

τ_{j}(A_{r}(−∞), ψ_{r}(−∞)) − τ_{j}(A_{r}(+∞), ψ_{r}(+∞))
= ∫_{Y (r)} f_{r}(A_{r}(−∞) − A_{r}(+∞)) ∧ ∗μ_{j}(r).

As in Lemma 4.6 of [4], up to changing the connections within the same gauge class, we have an estimate

∥A_{r}(−∞) − A_{r}(+∞)∥_{L^{2}(Y (r))} ≤ C(P_{0}) Cr,

where the right hand side grows linearly in r like the volume V ol(Y (r)). This estimate follows from the uniform pointwise bound on the spinor, and the corresponding bound on the curvature. Thus, for all ϵ > 0 we can choose r ≥ r_{0} large enough so that we have a bound

| ∫_{Y (r)} f_{r}(A_{r}(−∞) − A_{r}(+∞)) ∧ ∗μ_{j}(r)|

≤ CC(P_{0})V ol(Y (r_{0})) + ϵ.

The estimate of the remaining term in the variation of CSD_{P} is analogous. Combining these estimates, we get a uniform bound on the energy for large enough r ≥ r_{0}, hence the estimates on the finite energy solutions (A_{r}, Ψ_{r}) follow as in the Lemmata 2.3 and 2.5. ⋄

5.2 Asymptotics

In order to adapt the results of Section 3 we need to study the asymptotics of finite energy solutions of the equations (56) on the manifold V × R, with the non-compact end T^{2} × [0, ∞) × R.

As in Section 3.2, we consider the ODE associated to the perturbed system (56) on V × R. We write (56) as

∂_{t}a − dh + ∗(∂_{s}a − df) = ∗i(ᾱβ + αβ̄) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} ∗ q_{j}

∂_{t}f − ∂_{s}h + ∗F_{a} = \frac{i}{2}(|α|^{2} − |β|^{2}) + ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} p_{j}

∂_{t}α + hα + i∂_{s}α + ifα + ¯∂_{a}^{∗}β − i ∑_{j=1}^{N} \frac{∂V}{∂ζ_{j}} ((−ν_{j}^{1} + iν_{j}^{2})β + iν_{j}^{0}α) = 0

∂_{t}β + hβ − i∂_{s}β − ifβ + ¯∂_{a}α + i ∑_{j=1}^{N} \frac{∂V}{∂ζ_{j}} ((ν_{j}^{1} + iν_{j}^{2})α − iν_{j}^{0}β) = 0,

where we use the notation introduced in [4],

∗_{3}P_{U} = ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} (p_{j} + q_{j} ∧ ds)

and

P_{V} = ∑_{i=1}^{K} \frac{∂V}{∂ζ_{i}} (ν_{i}^{1}dx + ν_{i}^{2}dy + ν_{i}^{0}ds),

with ∗μ_{j} = p_{j} + q_{j} ∧ ds, and ν_{i} = ν_{i}^{1}dx + ν_{i}^{2}dy + ν_{i}^{0}ds. In radial gauge we obtain

\begin{matrix}∂_{ρ}f = e^{2ρ} ∗ (F_{a} + \frac{i}{2}(|α|^{2} − |β|^{2})ω + P_{0}) \\ ∂_{ρ}a = ∗(∂_{θ}a − df + i(ᾱβ + αβ̄) + P_{1}) \\ ∂_{ρ}α = i(∂_{θ}α + fα + e^{ρ+iθ}(¯∂_{a}^{∗}β + P_{11}α + P_{12}β)) \\ ∂_{ρ}β = −i(∂_{θ}β + fβ + e^{ρ−iθ}(¯∂_{a}α + P_{21}α + P_{22}β), \end{matrix}  (59)

where we use the notation

\begin{matrix}P_{0}(s) = ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} p_{j}(s) \\ P_{1}^{0,1}(s) = ∑_{j=1}^{N} \frac{∂U}{∂τ_{j}} q_{j}^{0,1}(s) \\ P (s) = ( \begin{matrix}P_{11} & P_{12} \\ P_{21} & P_{22} \end{matrix} ) = ∑_{j=1}^{N} \frac{∂V}{∂ζ_{j}} ( \begin{matrix}iν_{j}^{0} & −ν_{j}^{1} + iν_{j}^{2} \\ ν_{j}^{1} + iν_{j}^{2} & −iν_{j}^{0} \end{matrix} ) \end{matrix}  (60)

as in [4]

The perturbation terms P_{0}(s), P_{1}(s) and P (s) depend on (a, f, α, β) through the variables

τ_{j} = ∫ (A − A_{0}) ∧ ∗μ_{j}

and

ζ_{j} = ∫ ⟨ν_{j}.ψ, ψ⟩dv

(cf. [4]). By the condition that the perturbation is chosen in the class P_{δ} we obtain the estimates

\begin{aligned}∥P_{0}(ρ)∥_{L^{2}(T^{2}×\{θ\})} ≤ & C(U, V)∥(a, f, h)∥_{L^{2}(T^{2}×\{θ\})} \\ & · ∫_{−∞}^{∞} exp(−δe^{ρ} cos θ)dρ \end{aligned}

\begin{aligned}∥P_{1}(ρ)∥_{L^{2}(T^{2}×\{θ\})} ≤ & C(U, V)∥(a, f, h)∥_{L^{2}(T^{2}×\{θ\})} \\ & · ∫_{−∞}^{∞} exp(−δe^{ρ} cos θ)dρ \end{aligned}

\begin{aligned}∥P (ρ) · (α, β)∥_{L^{2}(T^{2}×\{θ\})} ≤ & C(U, V)∥(α, β)∥_{L^{2}(T^{2}×\{θ\})} \\ & · ∫_{−∞}^{∞} exp(−δe^{ρ} cos θ)dρ \end{aligned}  (61)

after the change of coordinates s + it = e^{ρ+iθ}.

In order to study the asymptotics of the system (59) we proceed as in Section 3. We consider the linear system given by the uncoupled systems (10) and (26) for the linear ASD and Dirac equations. We add the perturbation terms coming from the terms (60) in polar coordinates (ρ, θ). We first study the asymptotic of this perturbed system and show that the finite energy solutions are still exponentially decaying in the radial direction, as the solutions of the original systems (10) and (26). We then proceed as in the remaining of Section 3, to prove that the full system (59) has finite energy solutions that are exponentially decaying in the radial direction.

The system given by (10) and (26) together with the perturbation terms (60) is also uncoupled in the curvature and Dirac part. We discuss the behavior of the Dirac part: the curvature part is completely analogous.

Following [9], §X, Section 8, we consider all the systems of ODE’s of the form (26), for all (n, l, k), with the additional terms coming from the Fourier transform of the term P (ρ) · (α, β). These may no longer be uncoupled as the original (26). We can write this perturbed system in the form

\begin{matrix}X′ = M^{−}X + P^{−}(X, Y) \\ Y′ = M^{+}Y + P^{+}(X, Y), \end{matrix}  (62)

where the variables X correspond to the eigenvectors of the systems (26) with eigenvalues λ = ±λ_{i}^{nlk}, as in (27) with Re(λ) < 0, and Y corresponds to the eigenvectors with Re(λ) > 0.

Let λ_{0} be the smallest absolute value of the eigenvalues with Re(λ) < 0. Consider a fixed μ with −λ_{0} < μ < 0. According to Theorem 8.1 and 8.3 of [9], §X, the finite energy solutions of (62) will be of the form

X(ρ) = e^{M^{−}(ρ−ρ_{0})}X_{0} + ∫_{ρ_{0}}^{ρ} e^{M^{−}(ρ−τ)}P^{−}(X(τ), Y_{0})dτ

(cf. (8.15) of [9], §X).

Thus, the asymptotic decay of solutions is governed by the decay of solutions (28) of (26). The asymptotics of the original system (59) are then obtained by successive approximation as in Section 3.

With these results in place, the remaining of Section 3 and Section 4 extend with minor changes. The hypotheses of Lemma 4.1 are now satisfied for a generic choice of the perturbation P ∈ P_{δ}. Similarly, we have guaranteed that the linearization D_{A′,Ψ′} is surjective.

6 The boundary operators in the exact triangle

In this section we derive conclusions about the boundary operators in the Floer complexes C_{∗}(Y_{1}), C_{∗}(Y, μ) and C_{(∗)}(Y_{0}, s_{k}), where μ is the surgery perturbation introduced in [4], and C_{(∗)}(Y_{0}, s_{k}) is the integer lift of the Z_{2k}- graded Floer complex on Y_{0} with the Spin^{c}-structure s_{k}, as described in [13], (cf. [4], [6]).

Consider the boundary operator ∂_{Y,μ} of the Floer complex C_{∗}(Y, μ). The results of [4] show that, as abelian groups, the C_{q}(Y, μ) can be decomposed as a direct sum

C_{q}(Y, μ) = C_{q}(Y_{1}) ⊕ C_{q}(Y_{0}),

where we write C_{∗}(Y_{0}) for the term ⊕_{k}C_{(∗)}(Y_{0}, s_{k}) for simplicity of notation. Thus, we can decompose the boundary operator ∂_{Y,μ} as

∂_{Y,μ} = ( \begin{matrix}∂_{Y,μ}^{00} & ∂_{Y,μ}^{01} \\ ∂_{Y,μ}^{10} & ∂_{Y,μ}^{11} \end{matrix} ) ,  (63)

with

∂_{Y,μ}^{ij} : C_{∗}(Y_{i}) → C_{∗}(Y_{j}).

Proposition 6.1 We have identifications

∂_{Y,μ}^{11} ≡ ∂_{Y_{1}}

and

∂_{Y,μ}^{00} ≡ ∂_{Y_{0}},

where ∂_{Y_{0}} = ⊕_{k}∂_{Y_{0},s_{k}}.

Proof. The surgery perturbation [4] (cf. [2]) which identifies the asymptotic values of solutions of the 3-dim Seiberg-Witten equations on Y with the asymptotic values on Y_{1} and Y_{0} produces a corresponding identification of the geometric limits of flow lines in χ_{0}(T^{2}, Y) with those in χ_{0}(T^{2}, Y_{1}) and χ_{0}(T^{2}, Y_{0}). This is illustrated, in the case of Y and Y_{1}, in Figure 5. The identification of these asymptotic values, together with the gluing theorem proved in the previous sections, provides the identification of the corresponding approximate solutions, constructed as in the previous section, and therefore of the corresponding finite energy flow lines. This gives the relations between the boundary operators stated in the Proposition. The identification of flow lines in Y_{0} × R with a fixed Spin^{c}-structure s_{k} is analogous. The identification of ∂_{Y,μ}^{00} with ∂_{Y_{0}} then follows by noticing that, if two critical points in M_{Y,μ} are connected by a flow line in Y × R, then the asymptotic flat connections in χ_{0}(T^{2}, Y) are connected by a path in M_{ν(K)}.

There is no connected path in M_{ν(K)} connecting the asymptotic values in χ_{0}(T^{2}, Y) of two elements of ∪_{k}M_{Y_{0}}(s_{k}) with different Spin^{c} structures. ⋄

Figure 5: The identification in H^{1}(T^{2}, iR) of the geometric limits of flow lines on Y_{1} and Y

Thus, we can derive the following Corollary.

Corollary 6.2 Suppose that the homology sphere Y satisfies the following diagonal hypothesis. (D) The boundary operator ∂_{Y,μ} has no component ∂_{Y,μ}^{01} or ∂_{Y,μ}^{10}.

Then, the exact sequence of abelian groups

0 → C_{∗}(Y_{1}) →^{j_{∗}} C_{∗,μ}(Y) →^{π_{∗}} ⊕_{k}C_{(∗)}(Y_{0}, s_{k}) → 0,  (64)

induced by the decomposition

M_{Y,μ} ≅ M_{Y_{1}} ∪ ⋃_{s_{k}} M_{Y_{0}}(s_{k}),

is an exact sequence of chain complexes.

Unfortunately, we do not have any geometric characterization of hypothesis (D). The hypothesis certainly would be satisfied if we could guarantee that the pointwise convergence of Lemma 3.1 extends uniformly to an infinite strip T^{2} × [−r_{0}, r_{0}] × R. Then, as proved in Proposition 3.8, every finite energy flow line (A_{r}, Ψ_{r}) on Y (r)×R would converge uniformly in the smooth topology on the infinite strip to a flat connection. In particular, as proved in Proposition 3.8, this would imply that the only non-empty moduli spaces of flow lines would connect critical points that break through the same asymptotic flat connection on T^{2}. This condition is obviously stronger than hypothesis (D), but we do not know of specific examples where we can guarantee the uniform estimate of Proposition 3.8.

In the general case, there will be non-trivial components ∂_{Y,μ}^{01} or ∂_{Y,μ}^{10}, hence the maps in (64) are not chain homomorphisms. In [12] we shall discuss how to construct different maps

C_{∗}(Y_{1}) →^{I_{∗}} C_{∗,μ}(Y)

and

C_{∗,μ}(Y) →^{L_{∗}} ⊕_{k}C_{(∗)}(Y_{0}, s_{k})

that are chain homomorphisms, and how to use them together with (64) to construct an exact sequence.

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Matilde Marcolli, Department of Mathematics, Massachusetts Institute of Technology, 2-275. Cambridge, MA 02139, USA matilde@math.mit.edu

Bai-Ling Wang, Department of Pure Mathematics, University of Adelaide, Adelaide SA 5005, Australia. bwang@maths.adelaide.edu.au