COLLAPSING AND DIRAC-TYPE OPERATORS

JOHN LOTT

Abstract. We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-space X̌ with X = X̌/G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of the square of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.

1. Introduction

In a previous paper we analyzed the limit of the spectrum of the differential form Laplacian on a manifold, under a collapse with bounded diameter and bounded sectional curvature [17]. In the present paper, we extend the analysis of [17] to geometric Dirac-type operators. As the present paper is a sequel to [17], we refer to the introduction of [17] for background information about collapsing with bounded curvature and its relation to analytic questions.

Let M be a connected closed oriented Riemannian manifold of dimension n > 0. If M is spin then we put G = Spin(n) and if M is not spin then we put G = SO(n). The spinortype fields that we consider are sections of a vector bundle E^{M} associated to a G-Clifford module V, the latter being in the sense of Definition 2 of Section 2. The ensuing Dirac-type operator D^{M} acts on sections of E^{M}. We will think of the spectrum σ(D^{M}) of D^{M} as a set of real numbers with multiplicities, corresponding to possible multiple eigenvalues. For simplicity, in this introduction we will sometimes refer to the Dirac-type operators as acting on spinors, even though the results are more general.

We first consider a collapse in which the limit space is a smooth Riemannian manifold. The model case is that of a Riemannian affine fiber bundle.

Definition 1. A Riemannian affine fiber bundle is a smooth fiber bundle π : M → B whose fiber Z is an infranilmanifold and whose structure group is reduced from Diff(Z) to Aff(Z), along with

• A horizontal distribution T^{H}M whose holonomy lies in Aff(Z),

• A family of vertical Riemannian metrics g^{T Z} which are parallel with respect to the flat affine connections on the fibers Z_{b} and

• A Riemannian metric g^{T B} on B.

Date: May 1, 2000.

Research supported by NSF grant DMS-9704633.

Given a Riemannian affine fiber bundle π : M → B, there is a Riemannian metric g^{T M} on M constructed from T^{H}M, g^{T Z} and g^{T B}. Let R^{M} denote the Riemann curvature tensor of (M, g^{T M}), let Π denote the second fundamental forms of the fibers {Z_{b}}_{b∈B} and let T ∈ Ω^{2}(M; T Z) be the curvature of T^{H}M. Given b ∈ B, there is a natural flat connection on E^{M} |_{Z_{b}} which is constructed using the affine structure of Z_{b}. We define a Clifford bundle E^{B} on B whose fiber over b ∈ B consists of the parallel sections of E^{M} |_{Z_{b}}. The operator D^{M} restricts to a first-order differential operator D^{B} on C^{∞}(B; E^{B}). If V happens to be the spinor module then we show that D^{B} is the “quantization” of a certain superconnection on B. For general V, there is an additional zeroth-order term in D^{B} which depends on Π and T.

We show that the spectrum of D^{M} coincides with that of D^{B} up to a high level, which depends on the maximum diameter diam(Z) of the fibers {Z_{b}}_{b∈B}.

Theorem 1. There are positive constants A, A′ and C which only depend on n and V such that if ∥ R^{Z} ∥_{∞} diam(Z)^{2} ≤ A′ then the intersection of σ(D^{M}) with the interval

[− ( A diam(Z)^{−2} − C (∥ R^{M} ∥_{∞} + ∥ Π ∥_{∞}^{2} + ∥ T ∥_{∞}^{2}))^{1/2} ,
(A diam(Z)^{−2} − C (∥ R^{M} ∥_{∞} + ∥ Π ∥_{∞}^{2} + ∥ T ∥_{∞}^{2}))^{1/2}] (1.1)

equals the intersection of σ(D^{B}) with (1.1).

If Z = S^{1}, Π = 0 and V is the spinor module then we recover some results of [1, Section 4]; see also [12, Theorem 1.5]. The proof of Theorem 1 follows the same strategy as the proof of the analogous [17, Theorem 1]. Consequently, in the proof of Theorem 1, we only indicate the changes that need to be made in the proof of [17, Theorem 1] and refer to [17] for details.

Given B, Cheeger, Fukaya and Gromov showed that under some curvature bounds, any Riemannian manifold M which is sufficiently Gromov-Hausdorff close to B can be well approximated by a Riemannian affine fiber bundle [11]. Using this fact, we show that the spectrum of D^{M} can be uniformly approximated by that of a certain first-order differential operator D^{B} on B, at least up to a high level which depends on the Gromov-Hausdorff distance between M and B.

Given ϵ > 0 and two collections of real numbers {a_{i}}_{i∈I} and {b_{j}}_{j∈J}, we say that {a_{i}}_{i∈I} and {b_{j}}_{j∈J} are ϵ-close if there is a bijection α : I → J such that for all i ∈ I, |b_{α(i)} − a_{i}| ≤ ϵ.

Theorem 2. Let B be a fixed smooth connected closed Riemannian manifold. Given n ∈ ℤ^{+}, take G ∈ {SO(n), Spin(n)} and let V be a G-Clifford module. Then for any ϵ > 0 and K > 0, there are positive constants A(B, n, V, ϵ, K), A′(B, n, V, ϵ, K), and C(B, n, V, ϵ, K) so that the following holds. Let M be an n-dimensional connected closed oriented Riemannian manifold with a G-structure such that ∥ R^{M} ∥_{∞} ≤ K and d_{GH}(M, B) ≤ A′. Then there are a Clifford module E^{B} on B and a certain first-order differential operator D^{B} on C^{∞}(B; E^{B}) such that

1. \{sinh^{−1} (\frac{λ}{\sqrt{2K}} ) : λ ∈ σ(D^{M}) and λ^{2} ≤ A d_{GH}(M, B)^{−2} − C\} is ϵ-close to a subset of

\{sinh^{−1} (\frac{λ}{\sqrt{2K}} ) : λ ∈ σ(D^{B})\}, and 2. \{sinh^{−1} (\frac{λ}{\sqrt{2K}} ) : λ ∈ σ(D^{B}) and λ^{2} ≤ A d_{GH}(M, B)^{−2} − C\} is ϵ-close to a subset of

\{sinh^{−1} (\frac{λ}{\sqrt{2K}} ) : λ ∈ σ(D^{M})\}.

The other results in this paper concern collapsing to a possibly-singular space. Let X be a limit space of a sequence {M_{i}}_{i=1}^{∞} of n-dimensional connected closed oriented Riemannian manifolds with uniformly bounded diameter and uniformly bounded sectional curvature. In general, X is not homeomorphic to a manifold. However, Fukaya showed that X is homeomorphic to X̌/G, where X̌ is a manifold and G is a compact Lie group which acts on X̌ [14]. This comes from writing M_{i} = P_{i}/G, where G = SO(n) and P_{i} is the oriented orthonormal frame bundle of M_{i}. There is a canonical Riemannian metric on P_{i}. Then {P_{i}}_{i=1}^{∞} has a subsequence which Gromov-Hausdorff converges to a manifold X̌. As the convergence argument can be done G-equivariantly, the corresponding subsequence of {M_{i}}_{i=1}^{∞} converges to X = X̌/G. In general, X̌ is a smooth manifold with a metric which is C^{1,α} regular for all α ∈ (0, 1).

In [17] we dealt with the limit of the spectra of the differential form Laplacians {△^{M_{i}}}_{i=1}^{∞} on the manifolds {M_{i}}_{i=1}^{∞}. We defined a limit operator △^{X} which acts on the “differential forms” on X, coupled to a superconnection. In order to make this precise, we defined the “differential forms” on X to be the G-basic differential forms on X̌. We constructed the corresponding differential form Laplacian △^{X} and showed that its spectrum described the limit of the spectra of {△^{M_{i}}}_{i=1}^{∞}. We refer to [17] for the precise statements.

In the case of geometric Dirac-type operators D^{M_{i}}, there is a fundamental problem in extending this approach. Namely, if X̌ is a spin manifold on which a compact Lie group G acts isometrically and preserving the spin structure then there does not seem to be a notion of G-basic spinors on X̌. In order to get around this problem, we take a different approach. For a given n-dimensional Riemannian spin manifold M, put G = Spin(n), let P be the principal Spin(n)-bundle of M and let V be the spinor module. One can identify the spinor fields on M with (C^{∞}(P) ⊗ V)^{G}, the G-invariant subspace of C^{∞}(P) ⊗ V. There are canonical horizontal vector fields {𝔜_{j}}_{j=1}^{n} on P and the Dirac operator takes the form D^{M} = − i ∑_{j=1}^{n} γ^{j} 𝔜_{j}. Furthermore, (D^{M})^{2} can be written in a particularly simple form. As in equation (4.2) below, when acting on (C^{∞}(P) ⊗ V)^{G}, (D^{M})^{2} becomes the scalar Laplacian on P (acting on V -valued functions) plus a zeroth-order term.

Following this viewpoint, it makes sense to define the limiting “spinor fields” on X to be the elements of (C^{∞}(X̌) ⊗ V )^{G}. We can then extend Theorem 1 to the setting of Gequivariant Riemannian affine fiber bundles. Namely, the limit operator D^{X} turns out to be a G-invariant first-order differential operator on C^{∞}(X̌) ⊗ V, transversally elliptic in the sense of Atiyah [2], which one then restricts to the G-invariant subspace (C^{∞}(X̌) ⊗ V )^{G}. In Theorem 6 below, we show that the analog of Theorem 1 holds, in which D^{B} is replaced by D^{X}.

Theorem 6 refers to a given G-equivariant Riemannian affine fiber bundle. In order to deal with arbitrary collapsing sequences, we use the aforementioned representation of (D^{M})^{2} as a Laplace-type operator on P. If {M_{i}}_{i=1}^{∞} is a sequence of n-dimensional Riemannian manifolds with uniformly bounded diameter and uniformly bounded sectional curvature then we show that after taking a subsequence, the spectra of {(D^{M_{i}})^{2}}_{i=1}^{∞} converge to the spectrum of a Laplace-type operator on a limit space. Let {λ_{k}(|D^{M}|)}_{k=1}^{∞} denote the eigenvalues of |D^{M}|, counted with multiplicity.

Theorem 3. Given n ∈ ℤ^{+} and G ∈ {SO(n), Spin(n)}, let {M_{i}}_{i=1}^{∞} be a sequence of connected closed oriented n-dimensional Riemannian manifolds with a G-structure. Let V be a G-Clifford module. Suppose that for some D, K > 0 and for each i ∈ ℤ^{+}, we have diam(M_{i}) ≤ D and ∥ R^{M_{i}} ∥_{∞} ≤ K. Then there are

1. A subsequence of {M_{i}}_{i=1}^{∞}, which we relabel as {M_{i}}_{i=1}^{∞},

2. A smooth closed G-manifold X̌ with a G-invariant Riemannian metric g^{T X̌} which is C^{1,α}-regular for all α ∈ (0, 1),

3. A positive G-invariant function χ ∈ C(X̌) with ∫X̌ χ dvol = 1 and

4. A G-invariant function V ∈ L^{∞}(X̌) ⊗ End(V) such that if △X̌ denotes the Laplacian on L^{2}(X̌, χ dvol) ⊗ V [13, (0.8)] and |D^{X}| denotes the operator \sqrt{△X̌ + V} acting on (L^{2}(X̌, χ dvol) ⊗ V )^{G} then for all k ∈ ℤ^{+},

lim_{i→∞} λ_{k} (|D^{M_{i}}|) = λ_{k} (|D^{X}|) .  (1.2)

In the special case of the signature operator, the proof of Theorem 3 is somewhat simpler than that of the analogous [17, Proposition 11], in that we essentially only have to deal with scalar Laplacians. However, [17, Proposition 11] gives more detailed information. In particular, it expresses the limit operator in terms of a basic flat degree-1 superconnection on X̌. This seems to be necessary in order to prove the results of [17] concerning small eigenvalues. Of course, one does not expect to have analogous results concerning the small eigenvalues of general geometric Dirac-type operators, as their zero-eigenvalues have no topological meaning.

As an application of Theorem 3, we give a characterization of manifolds which do not have a uniform upper bound on the k-th eigenvalue of |D^{M}|, in terms of diameter and sectional curvature.

Theorem 4. Let M be a connected closed oriented manifold with a G-structure. Let V be a G-Clifford module. Suppose that for some K > 0 and k ∈ ℤ^{+}, there is no uniform upper bound on λ_{k}(|D^{M}|) among Riemannian metrics on M with diam(M) = 1 and ∥ R^{M} ∥_{∞} ≤ K. Then M admits a possibly-singular fibration M → X whose generic fiber is an infranilmanifold Z such that the restriction of E^{M} to Z does not have any nonzero affine-parallel sections.

More precisely, the possibly-singular fibration M → X of Theorem 4 is the G-quotient of a G-equivariant Riemannian affine fiber bundle P → X̌. Theorem 4 is an analog of [17, Theorem 5.2]. A simple example of Theorem 4 comes from considering spinors on M = S^{1} × N, where N is a spin manifold and the spin structure on S^{1} is the one that does not admit a harmonic spinor. Upon shrinking the S^{1}-fiber, the eigenvalues of D_{M} go off to ± ∞. More generally, let π : M → B be an affine fiber bundle. Theorem 1 implies that if E^{M} |_{Z} does not have any nonzero affine-parallel sections then upon collapsing M to B as in [15, Section 6], the eigenvalues of D_{M} go off to ± ∞.

Finally, we state a result about the essential spectrum of the square of a geometric Diractype operator on a finite-volume manifold of pinched negative curvature, which is an analog of [18, Theorem 2]. Let M be a complete connected oriented n-dimensional Riemannian manifold with a G-structure. Suppose that M has finite volume and its sectional curvatures satisfy −b^{2} ≤ K ≤ −a^{2}, with 0 < a ≤ b. Let V be a G-Clifford module. Label the ends of M by I ∈ {1, …, B}. An end of M has a neighborhood U_{I} whose closure is diffeomorphic to [0, ∞) × Z_{I}, where the first coordinate is the Busemann function corresponding to a ray exiting the end, and Z_{I} is an infranilmanifold. Let E^{M} be the vector bundle on M associated to the pair (G, V) and let D^{M} be the corresponding Dirac-type operator. If U_{I} lies far enough out the end then for each s ∈ [0, ∞), C^{∞} ({s} × Z_{I}; E^{M} |_{{s}×Z_{I}}) decomposes as the direct sum of a finite-dimensional space E_{I,s}^{B}, consisting of “bounded energy” sections, and its orthogonal complement, consisting of “high energy” sections. The vector spaces {E_{I,s}^{B}}_{s∈[0,∞)} fit together to form a vector bundle E_{I}^{B} on [0, ∞). Let P_{0} be orthogonal pro-

jection from ⨁_{I=1}^{B} C^{∞} (\overline{U_{I}}; E^{M} |\overline{U_{I}}) to ⨁_{I=1}^{B} C^{∞} ([0, ∞); E_{I}^{B}). Let (D^{M})_{0}^{2} be the restriction

of (D^{M})^{2} to ⨁_{I=1}^{B} C^{∞} (\overline{U_{I}}; E^{M} |\overline{U_{I}}), say with Atiyah-Patodi-Singer boundary conditions.

Then P_{0} (D^{M})_{0}^{2} P_{0} is a second-order ordinary differential operator on ⨁_{I=1}^{B} C^{∞} ([0, ∞); E_{I}^{B}).

Theorem 5. The essential spectrum of (D^{M})^{2} is the same as that of P_{0} (D^{M})_{0}^{2} P_{0}.

Using Propositions 1 and 2, the proof of Theorem 5 is similar in structure to that of [18, Theorem 2], after one replaces the operator d+d^{∗} of [18] by the operator D^{M}. Consequently, we do not give the details here.

There is some intersection between Theorem 5 and the results of [4, Theorem 0.1], concerning the essential spectrum of D^{M} when n = 2 and under an additional curvature assumption, and [5, Theorem 1], concerning the essential spectrum of D^{M} when M is hyperbolic and V is the spinor module.

I thank the Max-Planck-Institut-Bonn for its hospitality while this research was performed.

2. Dirac-type Operators and Infranilmanifolds

Given n ∈ ℤ^{+}, let G be either SO(n) or Spin(n).

Definition 2. A G-Clifford module consists of a finite-dimensional Hermitian G-vector space V and a G-equivariant linear map γ : ℝ^{n} → End(V) such that γ(v)^{2} = |v|^{2} Id. and γ(v)^{∗} = γ(v).

Let M be a connected closed oriented smooth n-dimensional Riemannian manifold. Put G = Spin(n) or G = SO(n), according as to whether or not M is spin. If M is spin, fix a spin structure. Let P be the corresponding principal G-bundle, covering the oriented orthonormal frame bundle. Its topological isomorphism class is independent of the choice of Riemannian metric. Given the Riemannian metric, there is a canonical ℝ^{n}-valued 1-form θ on P, the soldering form.

With respect to the standard basis {e_{j}}_{j=1}^{n} of ℝ^{n}, we write γ^{j} = γ(e_{j}). We also take a basis {σ^{ab}}_{a,b=1}^{n} for the representation of the Lie algebra 𝔤 on V, so that (σ^{ab})^{∗} = − σ^{ab} and

[σ^{ab}, σ^{cd}] = δ^{ad} σ^{bc} − δ^{ac} σ^{bd} + δ^{bc} σ^{ad} − δ^{bd} σ^{ac}.  (2.1)

The G-equivariance of γ implies

[γ^{a}, σ^{bc}] = δ^{ab} γ^{c} − δ^{ac} γ^{b}.  (2.2)

Examples :

1. If G = Spin(n) and V is the spinor representation of G then σ^{ab} = \frac{1}{4} [γ^{a}, γ^{b}].

2. If G = SO(n) and V = Λ^{∗}(ℝ^{n}) ⊗_{ℝ} ℂ, let E^{j} and I^{j} denote exterior and interior multiplication by e^{j}, respectively. Put γ^{j} = i (E^{j} − I^{j}) and γ̂^{j} = E^{j} + I^{j}. Then σ^{ab} = \frac{1}{4} ([γ^{a}, γ^{b}] + [γ̂^{a}, γ̂^{b}]).

Put E^{M} = P ×_{G} V. The Dirac-type operator D^{M} acts on the space C^{∞}(M; E^{M}). As the topological vector space C^{∞}(M; E^{M}) is independent of any choice of Riemannian metric on M, it makes sense to compare Dirac-type operators for different Riemannian metrics on M; see [16, Section 2] for further discussion.

Let g^{T M} be the Riemannian metric on M. Let ω be the Levi-Civita connection on P. Let {e_{j}}_{j=1}^{n} be a local oriented orthonormal basis of T M, with dual basis {τ^{j}}_{j=1}^{n}. Then we can write ω locally as a matrix-valued 1-form ω_{b}^{a} = ∑_{j=1}^{n} ω_{bj}^{a}τ^{j}, and

D^{M} = − i ∑_{j=1}^{n} γ^{j} ∇_{e_{j}} = − i ∑_{j=1}^{n} γ^{j} (e_{j} + \frac{1}{2} ∑_{a,b=1}^{n} ω_{abj} σ^{ab}) .  (2.3)

We have the Bochner-type equation

(D^{M})^{2} = ∇^{∗}∇ − \frac{1}{8} ∑_{a,b,i,j=1}^{n} R_{abij}^{M} (γ^{i} γ^{j} − γ^{j} γ^{i}) σ^{ab}.  (2.4)

As the set of Riemannian metrics on M is an affine space modeled on a Fréchet space, it makes sense to talk about an analytic 1-parameter family {c(t)}_{t∈[0,1]} of metrics. Then for t ∈ [0, 1], ċ(t) is a symmetric 2-tensor on M. Let ∥ ċ(t) ∥_{c(t)} denote the norm of ċ(t) with respect to c(t), i.e.

∥ ċ(t) ∥_{c(t)} = sup_{v∈T M−0} \frac{|ċ(t)(v, v)|}{c(t)(v, v)}.  (2.5)

Put l(c) = ∫_{0}^{1} ∥ ċ(t) ∥_{c(t)} dt. We extend the definition of l(c) to piecewise-analytic families of metrics in the obvious way. Given K > 0, let M(M, K) be the set of Riemannian metrics on M with ∥ R^{M} ∥_{∞} ≤ K. Let d be the corresponding length metric on M(M, K), computed using piecewise-analytic paths in M(M, K). Let σ(D^{M}, g^{T M}) denote the spectrum of D^{M} as computed with g^{T M}, a discrete subset of ℝ which is counted with multiplicity.

Proposition 1. There is a constant C = C(n, V) > 0 such that for all K > 0 and g_{1}^{T M}, g_{2}^{T M} ∈ M(M, K),

\{sinh^{−1} (\frac{λ}{\sqrt{K}} ) : λ ∈ σ(D^{M}, g_{1}^{T M})\}  (2.6)

and

\{sinh^{−1} (\frac{λ}{\sqrt{K}} ) : λ ∈ σ(D^{M}, g_{2}^{T M})\}  (2.7)

are C d(g_{1}^{T M}, g_{2}^{T M})-close.

Proof. It is enough to show that there is a number C such that if {c(t)}_{t∈[0,1]} is an analytic 1parameter family of metrics contained in M(M, K) then \{sinh^{−1} (\frac{λ}{\sqrt{K}} ) : λ ∈ σ(D^{M}, c(0))\}

and \{sinh^{−1} (\frac{λ}{\sqrt{K}} ) : λ ∈ σ(D^{M}, c(1))\} are Cd(c(0), c(1))-close. By eigenvalue perturbation theory [19, Chapter XII], the subset ⋃_{t∈[0,1]}{t}×σ(D^{M}, c(t)) of ℝ^{2} is the union of the graphs of functions {λ_{j}(t)}_{j∈ℤ} which are analytic in t. Thus it is enough to show that for each j ∈ ℤ,

|sinh^{−1} (\frac{λ_{j}(1)}{\sqrt{K}}) − sinh^{−1} (\frac{λ_{j}(0)}{\sqrt{K}})| ≤ C l(c).  (2.8)

Let D(t) denote the Dirac-type operator constructed with the metric c(t). It is selfadjoint when acting on L^{2}(E^{M}, dvol(t)). In order to have all of the operators {D(t)}_{t∈[0,1]} acting on the same Hilbert space, define f(t) ∈ C^{∞}(M) by f(t) = \frac{dvol(t)}{dvol(0)}. Then the spectrum of D(t), acting on L^{2}(E^{M}, dvol(t)), is the same as the spectrum of the self-adjoint operator f(t)^{1/2} D(t) f(t)^{− 1/2} acting on L^{2}(E^{M}, dvol(0)). One can now compute \frac{dλ_{j}}{dt} using eigenvalue perturbation theory, as in [19, Chapter XII]. Let ψ_{j}(t) be a smoothly-varying unit eigenvector whose eigenvalue is λ_{j}(t). Define a quadratic form T (t) on T M by

T (t)(X, Y) = ⟨ψ_{j}, − i (γ(X)∇_{Y} ψ_{j} + γ(Y)∇_{X}ψ_{j})⟩ + ⟨− i (γ(X)∇_{Y} ψ_{j} + γ(Y)∇_{X}ψ_{j}), ψ_{j}⟩. (2.9)

Using the metric c(t) to convert the symmetric tensors ċ(t) and T (t) to self-adjoint sections of End(T M), one finds

\frac{dλ_{j}}{dt} = − \frac{1}{8} ∫_{M} Tr (ċ(t) T (t)) dvol(t).  (2.10)

(This equation was shown for the pure Dirac operator, by different means, in [10].) Then

|\frac{dλ_{j}}{dt} | ≤ const. ∥ ċ(t) ∥_{c(t)} ∫_{M} Tr(|T (t)|) dvol(t).  (2.11)

Letting {x_{i}}_{i=1}^{n} be an orthonormal basis of eigenvectors of T (t) at a point m ∈ M, we have Tr(|T (t)|) = ∑_{i=1}^{n} |T (t)(x_{i}, x_{i})|. Then from (2.9), we obtain

∫_{M} Tr(|T (t)|) dvol(t) ≤ const. (∫_{M} |∇ψ_{j}|^{2} dvol(t))^{1/2} .  (2.12)

From (2.4),

∫_{M} |∇ψ_{j}|^{2} dvol(t) ≤ λ_{j}^{2} + const. K.  (2.13)

In summary, from (2.11), (2.12) and (2.13), there is a positive constant C such that

|\frac{dλ_{j}}{dt} | ≤ C ∥ ċ(t) ∥_{c(t)} (λ_{j}^{2} + K)^{1/2} .  (2.14)

Integration gives equation (2.8). The proposition follows.

For some basic facts about infranilmanifolds, we refer to [17, Section 3]. Let N be a simply-connected connected nilpotent Lie group. Let Γ be a discrete subgroup of Aff(N) which acts freely and cocompactly on N. Put Z = Γ\N, an infranilmanifold. There is a canonical flat linear connection ∇^{aff} on T Z. Put Γ̂ = Γ ∩ N, a cocompact subgroup of N. There is a short exact sequence

1 ⟶ Γ̂ ⟶ Γ ⟶ F ⟶ 1,  (2.15)

with F a finite group. Put Ẑ = Γ̂\N, a nilmanifold which finitely covers Z with covering group F.

Let g^{T Z} be a Riemannian metric on Z which is parallel with respect to ∇^{aff}. Let us discuss the condition for Z to be spin. Suppose first that Z is spin. Choose a spin structure on Z. Fix the basepoint z_{0} = Γ e ∈ Z. As ∇^{aff} preserves g^{T Z}, its holonomy lies in SO(n). Hence ∇^{aff} lifts to a flat connection on the principal Spin(n)-bundle, which we also denote by ∇^{aff}. There is a corresponding holonomy representation Γ → Spin(n).

Conversely, suppose that we do not know a priori if Z is spin. Suppose that the affine holonomy Γ → F → SO(n) lifts to a homomorphism Γ → Spin(n). Naturally, the existence of this lifting is independent of the particular choice of g^{T Z}. Then there is a corresponding spin structure on Z with principal bundle Γ\(N × Spin(n)). The different spin structures on Z correspond to different lifts of Γ → SO(n) to Γ → Spin(n). These are labelled by H^{1}(Γ; ℤ_{2}) ≅ H^{1}(Z; ℤ_{2}). Note that there are examples of nonspin flat manifolds [3]. Also, even if Z is spin and has a fixed spin structure, the action of Aff(Z) on Z generally does not lift to the principal Spin(n)-bundle, as can be seen for the SL(n, ℤ)-action on Z = T^{n}.

Now let G be either SO(n) or Spin(n). Let V be a G-Clifford module. Suppose that Z has a G-structure. If G = SO(n) then we have the affine holonomy homomorphism ρ : Γ → SO(n). If G = Spin(n) then we have a given lift of it to ρ : Γ → Spin(n). In either case, there is an action of Γ on V coming from Γ →^{ρ} G → Aut(V). The vector bundle E^{Z} can now be written as E^{Z} = Γ\(N × V). We see that the vector space of sections of E^{Z} which are parallel with respect to ∇^{aff} is isomorphic to V^{Γ}, the subspace of V which is fixed by the action of Γ.

If V is the spinor representation of G = Spin(n) then let us consider the conditions for V^{Γ} to be nonzero. First, as the restriction of ρ : Γ → Spin(n) to Γ̂ maps Γ̂ to ±1, we must have ρ|Γ̂ = 1. Given this, the homomorphism ρ factors through a homomorphism F → Spin(n). Then we have V^{Γ} = V^{F}. This may be nonzero even if the homomorphism F → Spin(n) is nontrivial.

Returning to the case of general V, as g^{T Z} is parallel with respect to ∇^{aff}, the operator D^{Z} preserves the space V^{Γ} of affine-parallel sections of E^{Z}. Let D^{inv} be the restriction of D^{Z} to V^{Γ}.

Proposition 2. There are positive constants A, A′ and C depending only on dim(Z) and V such that if ∥ R^{Z} ∥_{∞} diam(Z)^{2} ≤ A′ then the spectrum σ(D^{Z}) of D^{Z} satisfies

σ(D^{Z}) ∩ [− (A diam(Z)^{−2} − C ∥ R^{Z} ∥_{∞})^{1/2} , (A diam(Z)^{−2} − C ∥ R^{Z} ∥_{∞})^{1/2}] =

σ(D^{inv}) ∩ [− (A diam(Z)^{−2} − C ∥ R^{Z} ∥_{∞})^{1/2} , (A diam(Z)^{−2} − C ∥ R^{Z} ∥_{∞})^{1/2}]. (2.16)

Proof. As D^{Z} is diagonal with respect to the orthogonal decomposition

C^{∞}(Z; E^{Z}) = V^{Γ} ⊕ (V^{Γ})^{⊥} ,  (2.17)

it is enough to show that there are constants A, A′ and C as in the statement of the proposition such that the eigenvalues of (D^{Z})^{2}|_{(V^{Γ})_{⊥}} are greater than A diam(Z)^{−2} − C ∥ R^{Z} ∥_{∞}. As in the proof of [17, Proposition 2], we can reduce to the case when F = {e}, i.e. Z is a nilmanifold Γ\N. Then

C^{∞}(Z; E^{Z}) ≅ (C^{∞}(N) ⊗ V)^{Γ} .  (2.18)

Using an orthonormal frame {e_{i}}_{i=1}^{dim(Z)} for the Lie algebra 𝔫 as in the proof of [17, Proposition 2], we can write

∇_{e_{i}}^{aff} = e_{i} ⊗ Id.  (2.19)

and

∇_{e_{i}}^{Z} = (e_{i} ⊗ Id.) + (Id. ⊗ \frac{1}{2} ∑_{a,b=1}^{dim(Z)} ω_{abi} σ^{ab}) .  (2.20)

The rest of the proof now proceeds as in that of [17, Proposition 2], to which we refer for details.

3. Collapsing to a Smooth Base

For background information about superconnections and their applications, we refer to [7]. Let M be a connected closed oriented Riemannian manifold which is the total space of a Riemannian submersion π : M → B. Suppose that M has a G^{M}-structure and that V^{M} is a G^{M}-Clifford module, as in Section 2. If G^{M} = SO(n), put G^{Z} = SO(dim(Z)) and G^{B} = SO(dim(B)). If G^{M} = Spin(n), put G^{Z} = Spin(dim(Z)) and G^{B} = Spin(dim(B)). As a fiber Z_{b} has a trivial normal bundle in M, it admits a G^{Z}-structure. Fixing an orientation of T_{b}B fixes the G^{Z}-structure of Z_{b}. Note, however, that B does not necessarily have a G^{B}-structure. For example, if M is oriented then B is not necessarily oriented, as is shown in the example of S^{1} ×_{ℤ_{2}} S^{2} → ℝP^{2}, where the generator of ℤ_{2} acts on S^{1} by complex conjugation and on S^{2} by the antipodal map. And if M is spin then B is not necessarily spin, as is shown in the example of S^{5} → ℂP^{2}. What is true is that if the vertical tangent bundle T Z, a vector bundle on M, has a G^{Z}-structure then B has a G^{B}-structure.

Put E^{M} = P ×_{G^{M}} V^{M}. There is a Clifford bundle C on B with the property that C^{∞}(B; C) ≅ C^{∞}(M; E^{M}) [7, Section 9.2]. If dim(Z) > 0 then dim(C) = ∞. To describe C more explicitly, let V^{M} = ⨁_{l∈L} V_{l}^{B} ⊗ V_{l}^{Z} be the decomposition of V_{M} into irreducible representations of G^{B} × G^{Z} ⊂ G^{M}.

Examples :

1. If G^{M} = Spin(n) and V^{M} is the spinor representation then V^{B} and V^{Z} are spinor representations.

2. If G^{M} = SO(n) and V^{M} = Λ^{∗}(ℝ^{n}) ⊗_{ℝ} ℂ then V^{B} = Λ^{∗}(ℝ^{dim(B)}) ⊗_{ℝ} ℂ and V^{Z} = Λ^{∗}(ℝ^{dim(Z)}) ⊗_{ℝ} ℂ.

Let U be a contractible open subset of B. Choose an orientation on U. For b ∈ U, let E_{b,l}^{Z} be the vector bundle on Z_{b} associated to the pair (G^{Z}, V_{l}^{Z}). Then E^{M} |_{Z_{b}} ≅ ⨁_{l∈L} V_{l}^{B} ⊗ E_{b,l}^{Z}. The vector bundles {E_{b,l}^{Z}}_{b∈U} are the fiberwise restrictions of a vector bundle E_{l}^{Z} on π^{−1}(U), a vertical “spinor” bundle. There is a pushforward vector bundle W_{l} on U whose fiber W_{l,b} over b ∈ U is C^{∞}(Z_{b}; E_{b,l}^{Z}). If dim(Z) > 0 then dim(W_{l}) = ∞. There are Hermitian inner products {h^{W_{l}}}_{l∈L} on {W_{l}}_{l∈L} induced from the vertical Riemannian metric g^{T Z}. Furthermore, there are Clifford bundles {C_{l}}_{l∈L} on U for which the fiber C_{l,b} of C_{l} over b ∈ U is isomorphic to V_{l}^{B} ⊗ W_{l,b}. By construction, C^{∞} (Z_{b}; E^{M} |_{Z_{b}}) ≅ ⨁_{l∈L} C_{l,b}. The Clifford bundles {C_{l}}_{l∈L} exist globally on B and C = ⨁_{l∈L} C_{l}. The Dirac-type operator D^{M} decomposes as D^{M} = ⨁_{l∈L} D_{l}^{M}, where D_{l}^{M} acts on C^{∞}(B; C_{l}).

In order to write D_{l}^{M} explicitly, let us recall the Bismut superconnection on W_{l}. We will deal with each l ∈ L separately and so we drop the subscript l for the moment. We use the notation of [9, Section III(c)] to describe the local geometry of the fiber bundle M → B, and the Einstein summation convention. Let ∇^{T Z} denote the Bismut connection on T Z [7, Proposition 10.2], which we extend to a connection on E_{l}^{Z}. The Bismut superconnection on W [7, Proposition 10.15] is of the form

A = D^{W} + ∇^{W} − \frac{1}{4} c(T).  (3.1)

Here D^{W} is the fiberwise Dirac-type operator and has the form

D^{W} = − i γ^{j} ∇_{e_{j}}^{T Z} = − i γ^{j} (e_{j} + \frac{1}{2} ω_{pqj} σ^{pq}) .  (3.2)

Next, ∇^{W} is a Hermitian connection on W given by

∇^{W} = τ^{α} (∇_{e_{α}}^{T Z} − \frac{1}{2} ω_{αjj}) = τ^{α} (e_{α} + \frac{1}{2} ω_{jkα} σ^{jk} − \frac{1}{2} ω_{αjj}) .  (3.3)

Finally,

c(T) = i ω_{αβj} γ^{j} τ^{α} τ^{β}.  (3.4)

The superconnection A can be “quantized” into an operator D^{A} on C^{∞}(B; V^{B} ⊗ W). Explicitly,

D^{A} = − i γ^{j} (e_{j} + \frac{1}{2} ω_{pqj} σ^{pq})

− i γ^{α} (e_{α} + \frac{1}{2} ω_{βγα} σ^{βγ} + \frac{1}{2} ω_{jkα} σ^{jk} − \frac{1}{2} ω_{αjj})

+ i \frac{1}{2} ω_{αβj} γ^{j} σ^{αβ}.  (3.5)

Let V ∈ End(C_{l}) be the self-adjoint operator given by

V = − i (ω_{αjk} γ^{k} σ^{αj} + \frac{1}{2} ω_{αjj} γ^{α} + ω_{αβj} (γ^{j} σ^{αβ} + γ^{α} σ^{jβ})) .  (3.6)

Then restoring the index l everywhere,

D_{l}^{M} = D^{A_{l}} + V_{l}.  (3.7)

Examples :

1. If G^{M} = Spin(n) and V^{M} is the spinor representation then V = 0.

2. If G^{M} = SO(n) and V^{M} = Λ^{∗}(ℝ^{n}) ⊗_{ℝ} ℂ then

V = − \frac{1}{4} i (ω_{αjk} γ^{k} [γ̂^{α}, γ̂^{j}] + ω_{αβj}(γ^{j} [γ̂^{α}, γ̂^{β}] + γ^{α} [γ̂^{j}, γ̂^{β}])) .  (3.8)

Now suppose that π : M → B is a Riemannian affine fiber bundle. Then E^{M} |_{Z_{b}} inherits a flat connection from the flat affine connections on {E_{b,l}^{Z}}_{l∈L}. Let E^{B} be the Clifford bundle on B whose fiber over b ∈ B is the space of parallel sections of E^{M} |_{Z_{b}}. Then D^{M} restricts to a first-order differential operator D^{B} on C^{∞}(B; E^{B}).

Given b ∈ U and l ∈ L, let W_{l,b}^{inv} be the finite-dimensional subspace of W_{l,b} consisting of affine-parallel elements of C^{∞}(Z_{b}; E_{b,l}^{Z}). From the discussion in Section 2, W_{l,b}^{inv} is isomorphic to (V_{l}^{Z})^{Γ}. The vector spaces W_{l,b}^{inv} fit together to form a finite-dimensional subbundle W_{l}^{inv} of W_{l}. There is a corresponding finite-dimensional Clifford subbundle C_{l}^{inv} of C_{l} whose fiber over b ∈ U is isomorphic to V_{l}^{B} ⊗ W_{l,b}^{inv}. Again, C_{l}^{inv} exists globally on B. Then E^{B} = ⨁_{l∈L} C_{l}^{inv}. Let D_{l}^{B} be the restriction of D_{l}^{M} to C^{∞}(B; C_{l}^{inv}). Then

D^{B} = ⨁_{l∈L} D_{l}^{B}.  (3.9)

The superconnection A_{l} restricts to an superconnection A_{l}^{inv} on W_{l}^{inv}, the endomorphism V_{l} restricts to an endomorphism of C_{l}^{inv} and D_{l}^{M} restricts to the first-order differential operator

D_{l}^{B} = D^{A_{l}^{inv}} + V_{l}^{inv}  (3.10)

on C^{∞}(B; C_{l}^{inv}).

Proof of Theorem 1 : The operator D_{l}^{M} is diagonal with respect to the orthogonal decomposition

C_{l} = C_{l}^{inv} ⊕ (C_{l}^{inv})^{⊥} .  (3.11)

Thus it suffices to show that there are constants A, A′ and C such that the spectrum of σ(D_{l}^{M}), when restricted to (C_{l}^{inv})^{⊥}, is disjoint from (1.1).

For simplicity, we drop the subscript l. Given η ∈ C^{∞} (B; (C^{inv})^{⊥}) ⊂ C^{∞}(M; E^{M}), it is enough to show that for suitable constants,

⟨D^{M}η, D^{M}η⟩ ≥ (const. diam(Z)^{−2} − const. (∥ R^{M} ∥_{∞} + ∥ Π ∥_{∞}^{2} + ∥ T ∥_{∞}^{2})) ⟨η, η⟩. (3.12)

Using (2.4), it is enough to show that

⟨∇^{M}η, ∇^{M}η⟩ ≥ (const. diam(Z)^{−2} − const. (∥ R^{M} ∥_{∞} + ∥ Π ∥_{∞}^{2} + ∥ T ∥_{∞}^{2})) ⟨η, η⟩. (3.13)

We can write ∇^{M} = ∇^{V} + ∇^{H}, where

∇^{V} : C^{∞}(M; E^{M}) → C^{∞}(M; T^{∗}Z ⊗ E^{M})  (3.14)

denotes covariant differentiation in the vertical direction and

∇^{H} : C^{∞}(M; E^{M}) → C^{∞}(M; π^{∗}T^{∗}B ⊗ E^{M})  (3.15)

denotes covariant differentiation in the horizontal direction. Then

⟨∇^{M}η, ∇^{M}η⟩ = ⟨∇^{V} η, ∇^{V} η⟩ + ⟨∇^{H}η, ∇^{H}η⟩
≥ ⟨∇^{V} η, ∇^{V} η⟩
= ∫_{B} ∫_{Z_{b}} |∇^{V} η|^{2}(z) dvol_{Z_{b}} dvol_{B}.  (3.16)

On a given fiber Z_{b}, we have

E^{M} |_{Z_{b}} ≅ V^{B} ⊗ E_{b}^{Z}.  (3.17)

Hence we can also use the Bismut connection ∇^{T Z} to vertically differentiate sections of E^{M}. That is, we can define

∇^{T Z} : C^{∞}(M; E^{M}) → C^{∞}(M; T^{∗}Z ⊗ E^{M}).  (3.18)

Explicitly, with respect to a local framing,

∇_{e_{j}}^{T Z} = e_{j} η + \frac{1}{2} ω_{pqj} σ^{pq} η  (3.19)

and

∇_{e_{j}}^{V} = e_{j} η + \frac{1}{2} ω_{pqj} σ^{pq} η + ω_{αkj} σ^{αk} η + \frac{1}{2} ω_{αβj} σ^{αβ} η.  (3.20)

Then from (3.16), (3.19) and (3.20),

⟨∇^{M}η, ∇^{M}η⟩ ≥ ∫_{B} [∫_{Z_{b}} |∇^{T Z}η|^{2}(z) − const. (∥ T_{b} ∥^{2} + ∥ Π_{b} ∥^{2}) |η(z)|^{2}] dvol_{Z_{b}} dvol_{B}. (3.21)

Thus it suffices to bound ∫_{Z_{b}} |∇^{T Z}η|^{2}(z) dvol_{Z_{b}} from below on a given fiber Z_{b} in terms of ⟨η, η⟩_{Z_{b}}, under the assumption that η ∈ (W_{b}^{inv})^{⊥}. Using the Gauss-Codazzi equation, we can estimate ∥ R^{Z_{b}} ∥_{∞} in terms of ∥ R^{M} ∥_{∞} and ∥ Π ∥_{∞}^{2}. Then the desired bound on ∫_{Z_{b}} |∇^{T Z}η|^{2}(z) dvol_{Z_{b}} follows from Proposition 2. □

Proof of Theorem 2 : Let g_{0}^{T M} denote the Riemannian metric on M. From Proposition 1, if a Riemannian metric g_{1}^{T M} on M is close to g_{0}^{T M} in (M(M, 2K), d) then applying the function x → sinh^{−1} (\frac{x}{\sqrt{2K}} ) to σ(D^{M}, g_{0}^{T M}) gives a collection of numbers which is close to that obtained by applying x → sinh^{−1} (\frac{x}{\sqrt{2K}} ) to σ(D^{M}, g_{1}^{T M}). We will use the geometric results of [11] to find a metric g_{2}^{T M} on M which is close to g_{0}^{T M} and to which we can apply Theorem 1.

First, as in [11, (2.4.1)], by the smoothing results of Abresch and others [11, Theorem 1.12], for any ϵ > 0 we can find metrics on M and B which are ϵ-close in the C^{1}-topology to the original metrics such that the new metrics satisfy ∥ ∇^{i}R ∥_{∞} ≤ A_{i}(n, ϵ) for some appropriate sequence {A_{i}(n, ϵ)}_{i=0}^{∞}. Let g_{1}^{T M} denote the new metric on M. In the proof of the smoothing result, such as using the Ricci flow [20, Proposition 2.5], one obtains an explicit smooth 1-parameter family of metrics on M in M(M, K′), for some K′ > K, going from g_{0}^{T M} to g_{1}^{T M}. We can approximate this family by a piecewise-analytic family. Hence one obtains an upper bound on d (g_{0}^{T M}, g_{1}^{T M}) in M(M, K′), for some K′ > K, which depends on K and is proportionate to ϵ. (Note that d is essentially the same as the C^{0}-metric on M(M, K′).) By rescaling, we may assume that ∥ R^{M} ∥_{∞} ≤ 1, ∥ R^{B} ∥_{∞} ≤ 1 and inj(B) ≥ 1. We now apply [11, Theorem 2.6], with B fixed. It implies that there are positive constants λ(n) and c(n, ϵ) so that if d_{GH}(M, B) ≤ λ(n) then there is a fibration f : M → B such that 1. diam (f^{−1}(b)) ≤ c(n, ϵ) d_{GH}(M, B).

2. f is a c(n, ϵ)-almost Riemannian submersion.

3. ∥ Π_{f^{−1}(b)} ∥_{∞} ≤ c(n, ϵ). As in [15], the Gauss-Codazzi equation, the curvature bound on M and the second fundamental form bound on f^{−1}(b) imply a uniform bound on {∥ R^{f^{−1}(b)} ∥_{∞}}_{b∈B}. Along with the

diameter bound on f^{−1}(b), this implies that if d_{GH}(M, B) is sufficiently small then f^{−1}(b) is almost flat.

From [11, Propositions 3.6 and 4.9], we can find another metric g_{2}^{T M} on M which is ϵclose to g_{1}^{T M} in the C^{1}-topology so that the fibration f : M → B gives M the structure of a Riemannian affine fiber bundle. Furthermore, by [11, Proposition 4.9], there is a sequence {A′_{i}(n, ϵ)}_{i=0}^{∞} so that we may assume that g_{1}^{T M} and g_{2}^{T M} are close in the sense that

∥ ∇^{i} (g_{1}^{T M} − g_{2}^{T M}) ∥_{∞} ≤ A′_{i}(n, ϵ) d_{GH}(M, B),  (3.22)

where the covariant derivative in (3.22) is that of the Levi-Civita connection of g_{2}^{T M}. Then we can interpolate linearly between g_{1}^{T M} and g_{2}^{T M} within M(M, K′′) for some K′′ > K′, and obtain an upper bound on d (g_{1}^{T M}, g_{2}^{T M}) in M(M, K′′) which is proportionate to ϵ. From [20, Theorem 2.1], we can take K′′ = 2K (or any number greater than K).

We now apply Theorem 1 to the Riemannian affine fiber bundle with metric g_{2}^{T M}. It remains to estimate the geometric terms appearing in (1.1). We have an estimate on ∥ Π ∥_{∞} as above. Applying O’Neill’s formula [8, (9.29)] to the Riemannian affine fiber bundle, we can estimate ∥ T ∥_{∞}^{2} in terms of ∥ R^{M} ∥_{∞} and ∥ R^{B} ∥_{∞}. Putting this together, the theorem follows. □

4. Collapsing to a Singular Base

Let 𝔭 : P → M be the principal G-bundle of Section 2. Let {𝔜_{j}}_{j=1}^{n} be the horizontal vector fields on P such that θ(𝔜_{j}) = e_{j}. Put D^{P} = − i ∑_{j=1}^{n} γ^{j} 𝔜_{j}, acting on C^{∞}(P) ⊗ V.

There is an isomorphism C^{∞}(M; E^{M}) ≅ (C^{∞}(P) ⊗ V)^{G}. Under this isomorphism, D^{M} ≅ D^{P} |_{(C^{∞}(P)⊗V)_{G}}. The Bochner-type equation (2.4) becomes

(D^{M})^{2} ≅ − ∑_{j=1}^{n} 𝔜_{j}^{2} − \frac{1}{8} ∑_{a,b,i,j=1}^{n}(𝔭^{∗}R^{M})_{abij} (γ^{i} γ^{j} − γ^{j} γ^{i}) σ^{ab}  (4.1)

when acting on (C^{∞}(P) ⊗ V)^{G}.

Let {x_{a}}_{a=1}^{dim(G)} be a basis for the Lie algebra 𝔤 which is orthonormal with respect to the negative of the Killing form. Let {𝔜_{a}}_{a=1}^{dim(G)} be the corresponding vector fields on P. Then − ∑_{a=1}^{dim(G)} 𝔜_{a}^{2} acts on (C^{∞}(P) ⊗ V)^{G} as c_{V} ∈ ℝ, the Casimir of the G-module V. Give P the Riemannian metric g^{T P} with the property that {𝔜_{j}, 𝔜_{a}} forms an orthonormal basis of vector fields. Let △^{P} denote the corresponding (nonnegative) scalar Laplacian on P, extended to act on C^{∞}(P) ⊗ V. Then when acting on (C^{∞}(P) ⊗ V)^{G}, equation (4.1) is equivalent to

(D^{M})^{2} ≅ △^{P} − \frac{1}{8} ∑_{a,b,i,j=1}^{n}(𝔭^{∗}R^{M})_{abij} (γ^{i} γ^{j} − γ^{j} γ^{i}) σ^{ab} − c_{V} Id.  (4.2)

Definition 3. A G-equivariant Riemannian affine fiber bundle structure on P consists of a Riemannian affine fiber bundle structure π̌ : P → X̌ which is G-equivariant.

Given a G-equivariant Riemannian affine fiber bundle, let Ž be the fiber of π̌ : P → X̌, an infranilmanifold. For collapsing purposes it suffices to take Ž to be a nilmanifold Γ\N [11, (7.2)]. We assume hereafter that this is the case. Put X = X̌/G, a possibly singular space. As N acts isometrically in a neighborhood of a given fiber Ž and preserves the horizontal subspaces of P → M, it follows that the vector fields {𝔜_{j}}_{j=1}^{n} are projectable with respect to π̌ and push forward to vector fields {X_{j}}_{j=1}^{n} on X̌. Put DX̌ = − i ∑_{j=1}^{n} γ^{j} X_{j}, acting on C^{∞}(X̌) ⊗ V. Let v ∈ C^{∞}(X̌) be given by v(x̌) = vol(Žx̌). We give C^{∞}(X̌) ⊗ V the weighted L^{2}-inner product with respect to the weight function v.

We recall that there is a notion of a pseudodifferential operator being transversally elliptic with respect to the action of a Lie group G [2, Definition 1.3].

Lemma 1. DX̌ is transversally elliptic on X̌.

Proof. Let s(DX̌) ∈ C^{∞}(T^{∗}X̌) ⊗ End(V) denote the symbol of DX̌. Suppose that ξ ∈ T ∗ ; x̌X̌ satisfies ξ(v̌) = 0 for all v̌ ∈ Tx̌X̌ which lie in the image of the representation of 𝔤 by vector fields on X̌. Then if p ∈ π̌^{−1}(x̌), we have that (π̌^{∗}ξ)(r) = 0 for all r ∈ T_{p}P which lie in the image of the representation of 𝔤 by vector fields on P. In other words, π̌^{∗}ξ is horizontal. Suppose in addition that s(DX̌)(ξ) = 0. Then s(D^{P})(π̌^{∗}ξ) = 0. As D^{P} is horizontally elliptic, it follows that π̌^{∗}ξ = 0. Thus ξ = 0, which proves the lemma.

Definition 4. For notation, write C^{∞}(X; E^{X}) = (C^{∞}(X̌) ⊗ V )^{G}. Let D^{X} be the restriction of DX̌ to C^{∞}(X; E^{X}).

It will follow from the proof of the next theorem that D^{X} is self-adjoint on the Hilbert space completion of C^{∞}(X; E^{X}) with respect to the (weighted) inner product. As DX̌ is transversally elliptic, it follows that D^{X} has a discrete spectrum [2, Proof of Theorem 2.2].

Let Π̌ denote the second fundamental forms of the fibers {Žx̌}_{x̌∈X̌}. Let Ť ∈ Ω^{2}(P; T Ž) be the curvature of the horizontal distribution on the affine fiber bundle P → X̌.

Theorem 6. There are positive constants A, A′ and C which only depend on n and V such that if ∥ RŽ ∥_{∞} diam(Ž)^{2} ≤ A′ then the intersection of σ(D^{M}) with

[− ( A diam(Ž)^{−2} − C (1 + ∥ R^{M} ∥_{∞} + ∥ Π̌ ∥_{∞}^{2} + ∥ Ť ∥_{∞}^{2}))^{1/2} ,

(A diam(Ž)^{−2} − C (1 + ∥ R^{M} ∥_{∞} + ∥ Π̌ ∥_{∞}^{2} + ∥ Ť ∥_{∞}^{2}))^{1/2}] (4.3)

equals the intersection of σ(D^{X}) with (4.3).

Proof. Let us write

C^{∞}(P) ⊗ V = (C^{∞}(X̌) ⊗ V ) ⊕ (C^{∞}(X̌) ⊗ V )^{⊥} ,  (4.4)

where we think of C^{∞}(X̌) ⊗ V as the elements of C^{∞}(P) ⊗ V which are constant along the fibers of the fiber bundle π̌ : P → X̌. Taking G-invariant subspaces, we have an orthogonal decomposition

C^{∞}(M; E^{M}) = C^{∞}(X; E^{X}) ⊕ (C^{∞}(X; E^{X}))^{⊥} ,  (4.5)

with respect to which D^{M} decomposes as

D^{M} = D^{X} ⊕ D^{M} |_{(C^{∞}(X;E^{X}))_{⊥}}.  (4.6)

As in the proof of Theorem 1, it suffices to obtain a lower bound on the spectrum of (D^{M})^{2}|_{(C^{∞}(X;E^{X}))_{⊥}}. As (C^{∞}(X; E^{X}))^{⊥} ⊂ (C^{∞}(X̌) ⊗ V )^{⊥}, using (4.2) it suffices to obtain a lower bound on the spectrum of △^{P} |_{(C^{∞}(X̌)⊗V)}_{⊥}. This follows from the arguments of the

proof of Theorem 1, using the fact that ∥ R^{P} ∥_{∞} ≤ const. (1 + ∥ R^{M} ∥_{∞}). We omit the details. In fact, it is somewhat easier than the proof of Theorem 1, since we are now only dealing with the scalar Laplacian and so can replace Proposition 2 by standard eigenvalue estimates (which just involve a lower Ricci curvature bound); see [6] and references therein.

Proof of Theorem 3 : Everything in the proof will be done in a G-equivariant way, so we may omit to mention this explicitly. Let P_{i} be the principal G-bundle of M_{i}, equipped with a Riemannian metric as in the beginning of the section. From the G-equivariant version of Gromov’s compactness theorem, we obtain a subsequence {P_{i}}_{i=1}^{∞} which converges in the equivariant Gromov-Hausdorff topology to a G-Riemannian manifold (X̌, g^{T X̌}) with a C^{1,α}-regular metric. As in [13, Section 3], the measure χ dvolX̌ is a weak-∗ limit point of the pushforwards of the normalized Riemannian measures on {P_{i}}_{i=1}^{∞}. As in [13, p. 535], after smoothing we may assume that we have G-equivariant Riemannian affine fiber bundles π̌_{i} : P′_{i} → X̌_{i}, with G acting freely on P′_{i}, along with G-diffeomorphisms ϕ̌_{i} : P_{i} → P′_{i} and Φ_{i} : X̌ → X̌_{i}. Put M′_{i} = P′_{i}/G. Then ϕ̌_{i} descends to a diffeomorphism ϕ_{i} : M_{i} → M′_{i} and we may also assume, as in the proof of Theorem 2, that

1. ϕ_{i}^{∗}g^{T M′_{i}} ∈ M(M_{i}, const. K),

2. d(ϕ_{i}^{∗}g^{T M′_{i}}, g^{T M_{i}}) ≤ 2^{− i} in M(M_{i}, const. K) and

3. lim_{i→∞} Φ_{i}^{∗}g^{T X̌_{i}} = g^{T X̌} in the C^{1,α}-topology. Using Proposition 1, we can effectively replace M_{i} by M′_{i} for the purposes of the argument. For simplicity, we relabel M′_{i} as M_{i} and P′_{i} as P_{i}. For the purposes of the limiting argument, using Theorem 6 and (4.2), we may replace the spectrum of |D^{M_{i}}| by the spectrum of the operator |D^{X_{i}}| ≡ \sqrt{△X̌_{i} + V_{i}} acting on C^{∞}(X_{i}, E^{X_{i}}) = (C^{∞}(X̌_{i}) ⊗ V )^{G}, where V_{i} is the restriction of

− \frac{1}{8} ∑_{a,b,i,j=1}^{n}(𝔭^{∗}R^{M_{i}})_{abij} (γ^{i} γ^{j} − γ^{j} γ^{i}) σ^{ab} − c_{V} Id.  (4.7)

to the elements of (C^{∞}(P_{i}) ⊗ V)^{G} which are constant along the fibers of π̌_{i} : P_{i} → X̌_{i}, i.e. to C^{∞}(X_{i}, E^{X_{i}}).

From the curvature bound, we have a uniform bound on {∥ V_{i} ∥_{∞}}_{i=1}^{∞}. Using the weak-∗ compactness of the unit ball, let V be a weak-∗ limit point of {Φ_{i}^{∗}V_{i}}_{i=1}^{∞} in L^{∞}(X̌)⊗End(V) = (L^{1}(X̌) ⊗ End(V))^{∗}. We claim that with this choice of X̌, χ and V, equation (1.2) holds.

To see this, we use the minimax characterization of eigenvalues as in [13, Section 5]. Using the diffeomorphisms {Φ_{i}}_{i=1}^{∞}, we identify each X̌_{i} with X̌. We denote by ⟨·, ·⟩_{X_{i}} an L^{2}-inner product constructed using Φ_{i}^{∗}g^{T X̌_{i}} and the weight function (π̌_{i})_{∗}(dvol_{P_{i}})/ ∫X̌_{i}(π̌_{i})_{∗}(dvol_{P_{i}}). We denote by ⟨·, ·⟩_{X} an L^{2}-inner product constructed using g^{T X̌} and the weight function χ dvolX̌. As △X̌ has a compact resolvent, it follows that |D^{X}|^{2} has a compact resolvent. Then

λ_{k}(|D^{X}|)^{2} = inf_{W} sup_{ψ∈W −0} \frac{⟨dψ, dψ⟩_{X} + ⟨ψ, Vψ⟩_{X}}{⟨ψ, ψ⟩_{X}} ,  (4.8)

where W ranges over the k-dimensional subspaces of the Sobolev space H^{1}(X; E^{X}). Given ϵ > 0, let W_{∞} be a k-dimensional subspace such that

sup_{ψ∈W_{∞}−0} \frac{⟨dψ, dψ⟩_{X} + ⟨ψ, Vψ⟩_{X}}{⟨ψ, ψ⟩_{X}} ≤ λ_{k}(|D^{X}|)^{2} + ϵ.  (4.9)

As ψ ⊗ ψ^{∗} lies in the finite-dimensional subspace W_{∞} ⊗ W_{∞}^{∗} of L^{1}(X̌) ⊗ End(V), it follows that

lim_{i→∞}⟨ψ, V_{i}ψ⟩_{X} = ⟨ψ, Vψ⟩_{X}  (4.10)

uniformly on {ψ ∈ W_{∞} : ⟨ψ, ψ⟩_{X} = 1}. Then

lim_{i→∞} sup_{ψ∈W_{∞}−0} \frac{⟨dψ, dψ⟩_{X_{i}} + ⟨ψ, V_{i}ψ⟩_{X_{i}}}{⟨ψ, ψ⟩_{X_{i}}} = sup_{ψ∈W_{∞}−0} \frac{⟨dψ, dψ⟩_{X} + ⟨ψ, Vψ⟩_{X}}{⟨ψ, ψ⟩_{X}} . (4.11)

As

λ_{k}(|D^{X_{i}}|)^{2} = inf_{W} sup_{ψ∈W −0} \frac{⟨dψ, dψ⟩_{X_{i}} + ⟨ψ, V_{i}ψ⟩_{X_{i}}}{⟨ψ, ψ⟩_{X_{i}}} ,  (4.12)

it follows that

lim sup_{i→∞} λ_{k}(|D^{X_{i}}|) ≤ λ_{k}(|D^{X}|).  (4.13)

We now show that

lim inf_{i→∞} λ_{k}(|D^{X_{i}}|) ≥ λ_{k}(|D^{X}|).  (4.14)

Along with (4.13), this will prove the theorem. Suppose that (4.14) is not true. Then there is some ϵ > 0 and some infinite subsequence of {M_{i}}_{i=1}^{∞}, which we relabel as {M_{i}}_{i=1}^{∞}, such that for all i ∈ ℤ^{+},

λ_{k}(|D^{X_{i}}|)^{2} ≤ λ_{k}(|D^{X}|)^{2} − 2 ϵ.  (4.15)

For each i ∈ ℤ^{+}, let W_{i} be a k-dimensional subspace of H^{1}(X; E^{X}) such that

sup_{ψ∈W_{i}−0} \frac{⟨dψ, dψ⟩_{X_{i}} + ⟨ψ, V_{i}ψ⟩_{X_{i}}}{⟨ψ, ψ⟩_{X_{i}}} ≤ λ_{k}(|D^{X_{i}}|)^{2} + ϵ.  (4.16)

Let {f_{i,j}}_{j=1}^{k} be a basis for W_{i} which is orthonormal with respect to ⟨·, ·⟩_{X}. Then for a given j, the sequence {f_{i,j}}_{i=1}^{∞} is bounded in H^{1}(X; E^{X}). After taking a subsequence, which we relabel as {f_{i,j}}_{i=1}^{∞}, we can assume that {f_{i,j}}_{i=1}^{∞} converges weakly in H^{1}(X; E^{X}) to some f_{∞,j}. Doing this successively for j ∈ {1, …, k}, we can assume that for each j, lim_{i→∞} f_{i,j} = f_{∞,j} weakly in H^{1}(X; E^{X}). Then from the compactness of the embedding H^{1}(X; E^{X}) → L^{2}(X; E^{X}), we have strong convergence in L^{2}(X; E^{X}). In particular, {f_{∞,j}}_{j=1}^{k} are orthonormal. Put W_{∞} = span(f_{∞,1}, …, f_{∞,k}).

If w_{∞} = ∑_{j=1}^{k} c_{j} f_{∞,j} is a nonzero element of W_{∞}, put w_{i} = ∑_{j=1}^{k} c_{j} f_{i,j}. Then {w_{i}}_{i=1}^{∞} converges weakly to w_{∞} in H^{1}(X; E^{X}) and hence converges strongly to w_{∞} in L^{2}(X; E^{X}). From a general result about weak limits, we have

⟨w_{∞}, w_{∞}⟩_{H^{1}} ≤ lim sup_{i→∞} ⟨w_{i}, w_{i}⟩_{H^{1}}.  (4.17)

Along with the L^{2}-convergence of {w_{i}}_{i=1}^{∞} to w_{∞}, this implies that

⟨dw_{∞}, dw_{∞}⟩_{X} ≤ lim sup_{i→∞} ⟨dw_{i}, dw_{i}⟩_{X_{i}}.  (4.18)

As w_{i} ⊗ w_{i}^{∗} converges in L^{1}(X̌) ⊗ End(E) to w_{∞} ⊗ w_{∞}^{∗}, we have

lim_{i→∞}⟨w_{i}, V_{i}w_{i}⟩_{X} = lim_{i→∞} (⟨w_{∞}, V_{i}w_{∞}⟩_{X} + (⟨w_{i}, V_{i}w_{i}⟩_{X} − ⟨w_{∞}, V_{i}w_{∞}⟩_{X}))
= ⟨w_{∞}, Vw_{∞}⟩_{X}.  (4.19)

Then

sup_{ψ∈W_{∞}−0} \frac{⟨dψ, dψ⟩_{X} + ⟨ψ, Vψ⟩_{X}}{⟨ψ, ψ⟩_{X}} ≤ lim sup_{i→∞} sup_{ψ∈W_{i}−0} \frac{⟨dψ, dψ⟩_{X_{i}} + ⟨ψ, V_{i}ψ⟩_{X_{i}}}{⟨ψ, ψ⟩_{X_{i}}} . (4.20)

Thus from (4.15), (4.16) and (4.20),

inf_{W} sup_{ψ∈W −0} \frac{⟨dψ, dψ⟩_{X} + ⟨ψ, Vψ⟩_{X}}{⟨ψ, ψ⟩_{X}} ≤ λ_{k}(|D^{X}|)^{2} − ϵ,  (4.21)

which is a contradiction. This proves the theorem. □

Proof of Theorem 4 : Let {g_{i}^{T M}}_{i=1}^{∞} be a sequence of Riemannian metrics on M as in the statement of the theorem, with respect to which λ_{k}(|D^{M}|) goes to infinity. Let P be the principal G-bundle of M and let X̌ be the limit space of Theorem 3, a smooth manifold with a C^{1,α}-regular metric. As the limit space X = X̌/G has diameter 1, it has positive dimension. As in the proof of Theorem 3, after slightly smoothing the metric on X̌, there is a G-equivariant Riemannian affine fiber bundle π̌ : P → X̌ whose fiber is a nilmanifold Ž. Let x̌ be a point in a principal orbit for the G-action on X̌, with isotropy group H ⊂ G. Then H acts affinely on the nilmanifold fiber Žx̌. In particular, H is virtually abelian. The quotient Z = Žx̌/H is the generic fiber of the possibly-singular fiber bundle π : M → X, the G-quotient of π̌ : P → X̌. Then E^{M} |_{Z} = Žx̌ ×_{H} V. In particular, the vector space of affine-parallel sections of E^{M} |_{Z} is isomorphic to V^{H}. On the other hand, if C^{∞}(X; E^{X}) ≠ 0 then |D^{X}| has an infinite discrete spectrum. Theorem 3 now implies that C^{∞}(X; E^{X}) ≅ (C^{∞}(X̌) ⊗ V )^{G} must be the zero space. As the orbit x̌ · G has a neighborhood consisting of principal orbits, the restriction map from (C^{∞}(X̌) ⊗ V )^{G} to (C^{∞}(x̌ · G) ⊗ V)^{G} is surjective. However, (C^{∞}(x̌ · G) ⊗ V)^{G} is isomorphic to V^{H}. Thus V^{H} = 0. This proves the theorem. □

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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA

  E-mail address: lott@math.lsa.umich.edu