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Collapsing and Dirac-Type Operators

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 Y, with X = Y/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 a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.


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 [18]. In the present paper, we extend the analysis of [18] to geometric Dirac-type operators. As the present paper is a sequel to [18], we refer to the introduction of [18] 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.  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 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 [18,Theorem 1]. Consequently, in the proof of Theorem 1, we only indicate the changes that need to be made in the proof of [18,Theorem 1] and refer to [18] 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 ǫ > 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 ǫ-close if there is a bijection α : I → J such that for all i ∈ I, |b α(i) − a i | ≤ ǫ.
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 ǫ > 0 and K > 0, there are positive constants A(B, n, V, ǫ, K), A ′ (B, n, V, ǫ, K), and C(B, n, V, ǫ, 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 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 toX/G, whereX is a manifold and G is a compact Lie group which acts onX [15]. 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 .
has a subsequence which Gromov-Hausdorff converges to a manifoldX. As the convergence argument can be done G-equivariantly, the corresponding subsequence of converges to X =X/G. In general,X is a smooth manifold with a metric which is C 1,α regular for all α ∈ (0, 1).
In [18] we dealt with the limit of the spectra of the differential form Laplacians 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 onX. We constructed the corresponding differential form Laplacian △ X and showed that its spectrum described the limit of the spectra of . We refer to [18] 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, ifX 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 onX. 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 } n j=1 on P and the Dirac operator takes the form D M = − i n j=1 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 ∞ (X) ⊗ 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 ∞ (X) ⊗ V , transversally elliptic in the sense of Atiyah [2], which one then restricts to the G-invariant subspace C ∞ (X) ⊗ 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 .
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 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 In the special case of the signature operator, the proof of Theorem 3 is somewhat simpler than that of the analogous [18,Proposition 11], in that we essentially only have to deal with scalar Laplacians. However, [18,Proposition 11] gives more detailed information. In particular, it expresses the limit operator in terms of a basic flat degree-1 superconnection onX. This seems to be necessary in order to prove the results of [18] 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 →X. Theorem 4 is an analog of [18,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 [16,Section 6], the eigenvalues of D M go off to ± ∞.
Finally, we give a result about the essential spectrum of a geometric Dirac-type operator on a finite-volume manifold of pinched negative curvature, which is an analog of [19,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, . . . , N}. An end of M has a neighborhood U I whose closure is homeomorphic 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, ∞), poses as the direct sum of a finite-dimensional space E B I,s , consisting of "bounded energy" sections, and its orthogonal complement, consisting of "high energy" sections. The vector , say with Atiyah-Patodi-Singer boundary conditions.
The essential spectrum of D M is the same as that of P 0 D M end P 0 . 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.

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
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 } n j=1 of R n , we write γ j = γ(e j ). We also take a basis {σ ab } n a,b=1 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) Examples : 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 γ j = E j + I j . Then We have the Bochner-type equation 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 ofċ(t) with respect to c(t), i.e. .
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.
and 1] is an analytic 1parameter family of metrics contained in M(M, K) then sinh −1 λ [20,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, 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 . 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 (0)). One can now compute dλ j dt using eigenvalue perturbation theory, as in [20,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 Using the metric c(t) to convert the symmetric tensorsċ(t) and T (t) to self-adjoint sections of End(T M), one finds (2.10) (This equation was shown for the pure Dirac operator, by different means, in [10].) Then In summary, from (2.11), (2.12) and (2.13), there is a positive constant C such that Integration gives equation (2.8). The proposition follows.
For some basic facts about infranilmanifolds, we refer to [18,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 ∇ af f on T Z. Put Γ = Γ ∩ N, a cocompact subgroup of N. There is a short exact sequence with F a finite group. Put Z = Γ\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 ∇ af f . 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 ∇ af f preserves g T Z , its holonomy lies in SO(n).
Hence ∇ af f lifts to a flat connection on the principal Spin(n)-bundle, which we also denote by ∇ af f . 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 ∇ af f 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 Γ maps Γ to ±1, we must have ρ Γ = 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 ∇ af f , 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 (2.16) Proof. As D Z is diagonal with respect to the orthogonal decomposition 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 ( for the Lie algebra n as in the proof of [18, Proposition 2], we can write and The rest of the proof now proceeds as in that of [18,Proposition 2], to which we refer for details.

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 The vector bundles {E Z b,l } b∈U are the fiberwise restrictions of a vector bundle E Z l 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 Z b,l ). 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 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 M l , where D M l acts on C ∞ (B; C l ). In order to write D M l 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 Z l . The Bismut superconnection on W [7, Proposition 10.15] is of the form Here D W is the fiberwise Dirac-type operator and has the form Next, ∇ W is a Hermitian connection on W given by Finally, The superconnection A can be "quantized" into an operator D A on C ∞ (B; V B ⊗ W ). Explicitly, Let V ∈ End(C l ) be the self-adjoint operator given by  . 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 inv l,b be the finite-dimensional subspace of W l,b consisting of affine-parallel elements of C ∞ (Z b ; E Z b,l ). From the discussion in Section 2, W inv l,b is isomorphic to V Z l Γ . The vector spaces W inv l,b fit together to form a finite-dimensional subbundle W inv l of W l . There is a corresponding finite-dimensional Clifford subbundle C inv l of C l whose The superconnection A l restricts to an superconnection A inv l on W inv l , the endomorphism V l restricts to an endomorphism of C inv l and D M l restricts to the first-order differential operator Proof of Theorem 1 : The operator D M l is diagonal with respect to the orthogonal decomposition Thus it suffices to show that there are constants A, A ′ and C such that the spectrum of σ(D M l ), when restricted to (C inv l ) ⊥ , 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, Using (2.4), it is enough to show that We can write ∇ M = ∇ V + ∇ H , where denotes covariant differentiation in the vertical direction and denotes covariant differentiation in the horizontal direction. Then On a given fiber Z b , we have Hence we can also use the Bismut connection ∇ T Z to vertically differentiate sections of E M . That is, we can define Explicitly, with respect to a local framing, and Then from (3.16), (3.19) and (3.20), (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 inv b ) ⊥ . Using the Gauss-Codazzi equation, we can estimate R Z b ∞ in terms of R M ∞ and Π 2 ∞ . Then the desired bound on to σ(D M , g T M 0 ) gives a collection of numbers which is close to that obtained by applying to σ(D M , g T M 1 ). We will use the geometric results of [11] to find a metric g T M 2 on M which is close to g T M 0 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 ǫ > 0 we can find metrics on M and B which are ǫ-close in the C 1 -topology to the original metrics such that the new metrics satisfy ∇ i R ∞ ≤ A i (n, ǫ) for some ap- in M(M, K ′ ), for some K ′ > K, which depends on K and is proportionate to ǫ. (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, ǫ) so that if d GH (M, B) ≤ λ(n) then there is a fibration f : ≤ c(n, ǫ). As in [16], the Gauss-Codazzi equation, the curvature bound on M and the second fundamental form bound on f −1 (b) imply a uniform bound on are close in the sense that in M(M, K ′′ ) which is proportionate to ǫ. From [21, 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 T M 2 . 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.

Collapsing to a Singular Base
Let p : P → M be the principal G-bundle of Section 2. Let {Y j } n j=1 be the horizontal vector fields on P such that θ(Y j ) = e j . Put be a basis for the Lie algebra g which is orthonormal with respect to the negative of the Killing form. Let {Y a } dim(G) a=1 be the corresponding vector fields on P . Then − dim(G) a=1 Y 2 a 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 Definition 3. A G-equivariant Riemannian affine fiber bundle structure on P consists of a Riemannian affine fiber bundle structureπ : P →X which is G-equivariant. Given a G-equivariant Riemannian affine fiber bundle, letŽ be the fiber ofπ : P →X, an infranilmanifold. For collapsing purposes it suffices to takeŽ to be a nilmanifold Γ\N [11, (7.2)]. We assume hereafter that this is the case. Put X =X/G, a possibly singular space. As N acts isometrically in a neighborhood of a given fiberŽ and preserves the horizontal subspaces of P → M, it follows that the vector fields {Y j } n j=1 are projectable with respect toπ and push forward to vector fields {X j } n j=1 onX. Put DX = − i n j=1 γ j X j , acting on C ∞ (X) ⊗ V . Let v ∈ C ∞ (X) be given by v(x) = vol(Žx). We give C ∞ (X) ⊗ 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].
Proof. Let s(DX) ∈ C ∞ (T * X ) ⊗ End(V ) denote the symbol of DX. Suppose that ξ ∈ T * xX satisfies ξ(v) = 0 for allv ∈ TxX which lie in the image of the representation of g by vector fields onX. Then if p ∈π −1 (x), we have that (π * 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,π * ξ is horizontal. Suppose in addition that s(DX)(ξ) = 0. Then s(D P )(π * ξ) = 0. As D P is horizontally elliptic, it follows thatπ * ξ = 0. Thus ξ = 0, which proves the lemma. Definition 4. For notation, write C ∞ (X; E X ) = C ∞ (X) ⊗ V G . Let D X be the restriction of DX 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 DX is transversally elliptic, it follows that D X has a discrete spectrum [2, Proof of Theorem 2.2].
LetΠ denote the second fundamental forms of the fibers {Žx}x ∈X . LetŤ ∈ Ω 2 (P ; TŽ) be the curvature of the horizontal distribution on the affine fiber bundle P →X.
Proof. Let us write where we think of C ∞ (X) ⊗ V as the elements of C ∞ (P ) ⊗ V which are constant along the fibers of the fiber bundleπ : P →X. Taking G-invariant subspaces, we have an orthogonal decomposition with respect to which D M decomposes as 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 ∞ (X) ⊗ V ⊥ , using (4.2) it suffices to obtain a lower bound on the spectrum of △ P (C ∞ (X)⊗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 X , g TX with a C 1,α -regular metric. As in [14,Section 3], the measure χ dvolX is a weak- * limit point of the pushforwards of the normalized Riemannian measures on {P i } ∞ i=1 . As in [14, p. 535], after smoothing we may assume that we have G-equivariant Riemannian affine fiber bundlesπ i : i and we may also assume, as in the proof of Theorem 2, that const. K) and 3. lim i→∞ Φ * i g TX i = g TX 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 to the elements of (C ∞ (P i ) ⊗ V ) G which are constant along the fibers ofπ i : From the curvature bound, we have a uniform bound on Using the weak- * compactness of the unit ball, let V be a weak- * limit point of We claim that with this choice ofX, χ and V, equation (1.2) holds.
To see this, we use the minimax characterization of eigenvalues as in [14,Section 5]. Using the diffeomorphisms {Φ i } ∞ i=1 , we identify eachX i withX. We denote by ·, · X i an L 2 -inner product constructed using Φ * i g TX i and the weight function (π i ) * (dvol P i )/ X i (π i ) * (dvol P i ). We denote by ·, · X an L 2 -inner product constructed using g TX and the weight function χ dvolX. As △X has a compact resolvent, it follows that |D X | 2 has a compact resolvent. Then where W ranges over the k-dimensional subspaces of the Sobolev space H 1 (X; E X ). Given ǫ > 0, let W ∞ be a k-dimensional subspace such that uniformly on {ψ ∈ W ∞ : ψ, ψ X = 1}. Then it follows that We now show that (4.14) Along with (4.13), this will prove the theorem. Suppose that (4.14) is not true. Then there is some ǫ > 0 and some infinite subsequence of For each i ∈ Z + , let W i be a k-dimensional subspace of H 1 (X; E X ) such that Let {f i,j } k j=1 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, 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 dψ, dψ X + ψ, Vψ X ψ, ψ X ≤ λ k (|D X |) 2 − ǫ, (4.21) which is a contradiction. This proves the theorem.

Proof of Theorem 4 :
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 letX be the limit space of Theorem 3, a smooth manifold with a C 1,α -regular metric. As the limit space X =X/G has diameter 1, it has positive dimension. As in the proof of Theorem 3, after slightly smoothing the metric onX, there is a G-equivariant Riemannian affine fiber bundleπ : P →X whose fiber is a nilmanifoldŽ. Letx be a point in a principal orbit for the G-action onX, with isotropy group H ⊂ G. Then H acts affinely on the nilmanifold fiberŽx. In particular, H is virtually abelian. The quotient Z =Žx/H is the generic fiber of the possibly-singular fiber bundle π : M → X, the G-quotient ofπ : P →X. Then E M Z =Žx × 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 ∞ (X) ⊗ V G must be the zero space. As the orbitx · G has a neighborhood consisting of principal orbits, the restriction map from C ∞ (X) ⊗ V G to (C ∞ (x · G) ⊗ V ) G is surjective. However, (C ∞ (x · G) ⊗ V ) G is isomorphic to V H . Thus V H = 0. This proves the theorem.

Proof of Theorem 5
As the proof of Theorem 5 is similar to [19, Pf. of Theorem 2], we only indicate the structure of the proof and the necessary modifications to [19,Pf. of Theorem 2].