Eigenvalue estimates and differential form Laplacians on Alexandrov spaces

TL;DR

This paper establishes eigenvalue bounds for differential form Laplacians on Alexandrov spaces and constructs these operators, linking their kernels to intersection homology, thus extending spectral analysis to singular spaces.

Contribution

It introduces a method to estimate eigenvalues and construct Laplacians on Alexandrov spaces, connecting spectral properties with topological invariants.

Findings

Eigenvalue upper bounds for differential form Laplacians on Alexandrov spaces

Construction of differential form Laplacians on singular spaces

Identification of the Laplacian kernel with intersection homology groups

Abstract

We give upper bounds on the eigenvalues of the differential form Laplacian on a compact Riemannian manifold. The proof uses Alexandrov spaces with curvature bounded below. We also construct differential form Laplacians on Alexandrov spaces. Under a local biLipschitz assumption on the Alexandrov space, which is conjecturally always satisfied, we show that the differential form Laplacian has a compact resolvent. We identify its kernel with an intersection homology group.

Equations

$${\lambda }_k\le c(n,K{D}^2)\frac{k^2}{D^2}.$$

$${\lambda }_{k,p}\le C_1(n,K{D}^2,v{D}^{-n})\frac{k^{\frac{2}{n}}}{D^2}.$$

$$\widetilde{\text{mathchar}1117}{a}_{i}m{b}_i>\pi -\delta ,$$

$$\widetilde{\text{mathchar}1117}{a}_{i}m{b}_j>\frac{\pi }{2}-\delta ,$$

$${\lambda }_{k,p}\le C_2(n,K{S}^2)\frac{k^{\frac{2}{s}}}{S^2}.$$

$${\lambda }_{k,p}\le {\kappa }_n\frac{k^{\frac{2}{s}}}{S^2}.$$

$${\lambda }_{k,p}=\underset{V}{\inf }\underset{\omega \in V,\omega \ne 0}{\sup }\frac{|d\omega {|}_{L^2}^2+|d^{\ast }\omega {|}_{L^2}^2}{|\omega {|}_{L^2}^2},$$

$${\lambda }_{k,p}=\underset{V}{\inf }\underset{\omega \in V,\omega \ne 0}{\sup }\frac{|d\omega {|}_{L^2}^2}{|\omega {|}_{L^2}^2},$$

$$\rho ({\omega }^{\prime })=\sum _{i=1}^N{\varphi }_i\rho ({\omega }^{\prime })=\sum _{i=1}^N\sum _j\rho ({\varphi }_i{f}_0^{j}d{f}_1^j\dots d{f}_{n-1}^j)$$

$$\rho (d{\omega }^{\prime })=\sum _{i=1}^N(d{\varphi }_i\wedge \rho ({\omega }^{\prime })+{\varphi }_i\rho (d{\omega }^{\prime }))=\sum _{i=1}^N\sum _j\rho (d({\varphi }_i{f}_0^j)d{f}_1^j\dots d{f}_{n-1}^j).$$

$$d({\varphi }_i{f}_0^j)d({\eta }_i{f}_1^j)\dots d({\eta }_i{f}_{n-1}^j)=$$

$$((d{\varphi }_i)f_0^j+{\varphi }_{i}d{f}_0^j)\cdot ((d{\eta }_i)f_1^j+{\eta }_{i}d{f}_1^j)\dots ((d{\eta }_i)f_{n-1}^j+{\eta }_{i}d{f}_{n-1}^j)$$

$$\rho \left(d({\varphi }_i{f}_0^j)d({\eta }_i{f}_1^j)\dots d({\eta }_i{f}_{n-1}^j)\right)=$$

$$((d{\varphi }_i)\rho (f_0^j)+{\varphi }_i\rho (d{f}_0^j))\wedge ((d{\eta }_i)\rho (f_1^j)+{\eta }_i\rho (d{f}_1^j))\wedge \dots \wedge (d{\eta }_i\wedge \rho (f_{n-1}^j)+{\eta }_i\rho (d{f}_{n-1}^j))=$$

$$((d{\varphi }_i)\rho (f_0^j)+{\varphi }_i\rho (d{f}_0^j))\wedge \rho (d{f}_1^j)\wedge \dots \wedge \rho (d{f}_{n-1}^j)=\rho (d({\varphi }_i{f}_0^j)d{f}_1^j\dots d{f}_{n-1}^j)).$$

$$0={\int }_{X^{\ast }}\rho (d({\omega }^{\prime }\wedge \omega ))={\int }_{X^{\ast }}\rho (d{\omega }^{\prime })\wedge \rho (\omega ).$$

$$0=$$

$$=$$

$${\int }_{X^{\ast }}\left(d{\omega }^{\prime }\wedge \omega +(-1{)}^{n-p-1}{\omega }^{\prime }\wedge \eta \right)=0.$$

$${\int }_{X^{\ast }}d{\omega }^{\prime }\wedge d\omega =0$$

$$Q(\omega )={\int }_{X^{\ast }}⟨d\omega ,d\omega ⟩{\mathrm{dvol}}_{X^{\ast }}.$$

$$\mathrm{Dom}({△}_p)=\{\omega \in {\Omega }_{L^2,d}^p(X)/\overline{\mathrm{Im}(d)}:d\omega \in \mathrm{Dom}(d^{\ast })\}.$$

$${\Omega }_{L^2}^{\ast }(X)\cong \left({\Omega }_{L^2}^{\ast }(X)/\overline{\mathrm{Im}(d)}\right)\oplus \overline{\mathrm{Im}(d)}$$

$${\Omega }_{L^2}^{\ast }(X)/\overline{\mathrm{Im}(d)}\cong \left({\Omega }_{L^2}^{\ast }(X)/\mathrm{Ker}(d)\right)\oplus \left(\mathrm{Ker}(d)/\overline{\mathrm{Im}(d)}\right).$$

$${\Omega }_{L^2}^{\ast }(X)/\mathrm{Ker}(d)\cong {\left(\mathrm{Ker}(d)\right)}^{\perp }$$

$$\mathrm{Ker}(d)/\overline{\mathrm{Im}(d)}\cong \mathrm{Ker}(△),$$

$${\Omega }_{L^2}^{\ast }(X)\cong {\left(\mathrm{Ker}(d)\right)}^{\perp }\oplus \mathrm{Ker}({△}_{\ast })\oplus \overline{\mathrm{Im}(d)}.$$

$${\Omega }_{L^2}^{\ast }(X)/\overline{\mathrm{Im}(d)}\cong {\left(\mathrm{Ker}(d)\right)}^{\perp }\oplus \mathrm{Ker}({△}_{\ast }).$$

$$Q^{Hodge}(\omega )={\int }_X\left(⟨d\omega ,d\omega ⟩+⟨d^{\ast }\omega ,d^{\ast }\omega ⟩\right){\mathrm{dvol}}_X$$

$${\int }_{U^{\ast }}\left(d{\omega }^{\prime }\wedge \omega +(-1{)}^{n-p-1}{\omega }^{\prime }\wedge \eta \right)=0.$$

$${\int }_{U_i^{\ast }}\left(d({\sigma }_i{\omega }^{\prime })\wedge \omega +(-1{)}^{n-p-1}{\sigma }_i{\omega }^{\prime }\wedge d{\omega }_i\right)=0.$$

$${\int }_{U^{\ast }}\left(d{\omega }^{\prime }\wedge \omega +(-1{)}^{n-p-1}{\omega }^{\prime }\wedge \sum _{i=1}^N{\sigma }_{i}d{\omega }_i\right)=0.$$

$${\lambda }_{k,p}(X)=\underset{V_{k,p}}{\inf }\underset{\psi \in V_{k,p},\psi \ne 0}{\sup }\frac{\Vert d\psi {\Vert }^2}{\Vert \psi {\Vert }^2},$$

Full text

Research partially supported by NSF grant DMS-1510192

1. Introduction

For a closed connected Riemannian manifold, let $\lambda_{k}$ denote the $k^{th}$ positive eigenvalue of the (nonnegative) function Laplacian, counted with multiplicity. In 1975, S.-Y. Cheng proved the following upper bound on the eigenvalues.

Theorem 1.1**.**

(Cheng [7]) There is a function $c:{\mathbb{Z}}^{+}\times{\mathbb{R}}\rightarrow{\mathbb{R}}^{+}$ with the following property. Given $n\in{\mathbb{Z}}^{+}$, $D\in{\mathbb{R}}^{+}$ and $K\in{\mathbb{R}}$, if $M$ is an $n$-dimensional closed connected Riemannian manifold with diameter $D$, and $\operatorname{Ric}_{M}\geq Kg_{M}$, then

[TABLE]

We give extensions of Cheng’s result to the differential form Laplacian. Let $\lambda_{k,p}$ denote the $k^{th}$ positive eigenvalue of the Hodge Laplacian $dd^{*}+d^{*}d$ on $p$-forms, counted with multiplicity. First, we assume a lower volume bound.

Theorem 1.3**.**

There is a function $C_{1}:{\mathbb{Z}}^{+}\times{\mathbb{R}}\times{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}^{+}$ with the following property. Given $n\in{\mathbb{Z}}^{+}$, $D\in{\mathbb{R}}^{+}$, $K\in{\mathbb{R}}$ and $v\in{\mathbb{R}}^{+}$, if $M$ is an $n$-dimensional closed connected Riemannian manifold with diameter $D$, sectional curvatures bounded below by $K$ and volume bounded below by $v$, then for all $p\in[0,n]$ we have

[TABLE]

To remove the lower volume bound, we use the notion of a strainer [3, §5]. Given a complete Riemannian manifold (or Alexandrov space) with curvature bounded below by $K\in{\mathbb{R}}$, and an integer $s>0$, an $s$-strainer of quality $\delta$ and size $S$ at a point $m$ consists of points $\{a_{i},b_{i}\}_{i=1}^{s}$ with $d(m,a_{i})=d(m,b_{i})=S$ so that

[TABLE]

whenever $i\neq j$. Here $\tilde{\mathchar 1117\relax}$ is the comparison angle at $m$, relative to the model space of constant sectional curvature $K$.

Theorem 1.6**.**

There is a function $C_{2}:{\mathbb{Z}}^{+}\times{\mathbb{R}}\rightarrow{\mathbb{R}}^{+}$ with the following property. Given $n\in{\mathbb{Z}}^{+}$ and $K\in{\mathbb{R}}$, let $M$ be an $n$-dimensional closed connected Riemannian manifold with sectional curvatures bounded below by $K$. Suppose that there is some $m\in M$ with an $s$-strainer of quality $\frac{1}{10}$ and size $S$, where $1\leq s\leq n$. Then for all $p\in[0,s]$, we have

[TABLE]

Corollary 1.8**.**

Given $n\in{\mathbb{Z}}^{+}$, there is some $\kappa_{n}<\infty$ with the following property. Let $M$ be an $n$-dimensional closed connected Riemannian manifold with nonnegative sectional curvature. Suppose that there is some $m\in M$ with an $s$-strainer of quality $\frac{1}{10}$ and size $S$, where $1\leq s\leq n$. Then for all $p\in[0,s]$, we have

[TABLE]

Remark 1.10*.*

The choice of $\frac{1}{10}$ for the quality of the strainer is arbitrary. In the proof of Theorem 1.6 we actually get upper eigenvalue bounds for the Laplacian on $\Omega^{p}(M)/\operatorname{Ker}(d)$ when $p<s$, and for the Laplacian on $\operatorname{Im}(d)\subset\Omega^{s}(M)$.

Theorem 1.6 implies Theorem 1.1 under the stronger assumption of a lower sectional curvature bound, by taking $s=1$ and $S=\frac{D}{2}$. One would not expect to be able to control eigenvalues of the $p$-form Laplacian from a lower Ricci curvature bound if $p\notin\{0,1,n-1,n\}$.

Remark 1.11*.*

Theorem 1.3 actually follows from Theorem 1.6. Under the hypotheses of Theorem 1.3, after rescaling the diameter to be $1$, convergence theory implies that there is some point with an $n$-strainer of quality $\frac{1}{10}$ and a certain size.

Remark 1.12*.*

To get upper eigenvalue bounds for the $p$-form Laplacian, Theorem 1.6 has an assumption about the existence of a $p$-strainer at some point. The need for some such assumption can be seen by taking $M=X\times S^{N}$, where $X$ is a closed Riemannian manifold, and shrinking the $S^{N}$-factor. If $\dim(X)<p$ and $N>p$ then the eigenvalues of the $p$-form Laplacian on $M$ go to infinity. As the sphere shrinks, there is clearly no $p$-strainer on $M$ of quality $\frac{1}{10}$ whose size is uniformly bounded below.

Taking $s=\dim(X)$, this example also shows the sharpness of the exponent $\frac{2}{s}$ in (1.7), which corresponds to Weyl-type asymptotics on an $s$-dimensional manifold.

The constant $c$ in Theorem 1.1 can be made explicit. The constants $C_{1}$ and $C_{2}$ in Theorems 1.3 and 1.6 are not explicit. The reason is that the proofs of Theorems 1.3 and 1.6 are by a contradiction argument. One common feature of all the proofs is the use of a minmax argument on an appropriate class of test functions or test forms. To prove Theorem 1.1, Cheng transplanted functions from a model space, using the exponential map from a point. Dodziuk extended Cheng’s result, by transplanting differential forms from a model space, to prove an analog of Theorem 1.3 under the stronger assumptions of a double sided curvature bound and a lower bound on the injectivity radius [9]. We instead pullback differential forms from an Alexandrov space (with curvature bounded below) that arises in the contradiction argument.

This leads to the question of whether differential form Laplacians make sense for Alexandrov spaces. (The function Laplacian on an Alexandrov space was studied in [21, 31].) To see some of the issues involved, note that on a smooth Riemannian manifold, when written in local coordinates, the differential form Laplacian $dd^{*}+d^{*}d$ involves two derivatives of the metric tensor. An Alexandrov space has a dense open set with the structure of a Riemannian Lipschitz manifold, meaning in particular that there is a Riemannian metric whose components, in local coordinates, are in $L^{\infty}_{loc}$ [26]. Hence defining $dd^{*}+d^{*}d$ directly on an Alexandrov space does not look promising.

Instead of trying to directly define the differential form Laplacian as an operator, one could try to define the putative spectrum. On a smooth closed Riemannian manifold $M$, the minmax formula says that

[TABLE]

where $V$ ranges over $k$-dimensional subspaces of $\Omega^{p}(M)$. In local coordinates, $d^{*}\omega$ involves first derivatives of the metric tensor. On an Alexandrov space, one knows that the first derivatives of the metric components exist as measures [26], but this is not enough to make sense of (1.13).

To gain another derivative, we use the observation, essentially due to Cheeger and Dodziuk [9], that the minmax equation (1.13) takes a nicer form if we look instead at the Laplacian $\triangle_{*}$ on $\Omega^{*}(M)/\overline{\operatorname{Im}(d)}$. For this Laplacian, the minmax equation becomes

[TABLE]

where $V$ now ranges over $k$-dimensional subspaces of $\Omega^{p}(M)/\overline{\operatorname{Im}(d)}$. The right-hand side of (1.14) does not involve any derivatives of the metric tensor. Using the Hodge decomposition and the isomorphism $\Omega^{*}(M)/\overline{\operatorname{Im}(d)}\cong\operatorname{Ker}(dd^{*}+d^{*}d)\oplus\overline{\operatorname{Im}(d^{*})}$, the spectrum of $\triangle_{*}$ is the same as the spectrum of $dd^{*}+d^{*}d$, with the multiplicities related by a factor of at most two.

For this reason, in making sense of a differential form Laplacian on an Alexandrov space, we only consider an analog of the Laplacian $\triangle_{*}$ on $\Omega^{*}(M)/\overline{\operatorname{Im}(d)}$.

Theorem 1.15**.**

If $X$ is a compact Alexandrov space then there is a well-defined nonnegative self-adjoint differential form Laplacian $\triangle_{*}$. When $X$ is a smooth Riemannian manifold $M$ with (possibly empty) convex boundary, the operator $\triangle_{*}$ becomes the usual Hodge Laplacian on $\Omega^{*}(M)/\overline{\operatorname{Im}(d)}$ with relative (Dirichlet) boundary conditions.

We prove Theorem 1.3 in the generality of compact Alexandrov spaces. The use of $\Omega^{*}/\operatorname{Ker}(d)$ is key in proving Theorems 1.3 and 1.6. Their proofs do not need the existence of the differential form Laplacian on the limit space, but rather the existence of differential forms.

We construct $\triangle_{*}$ more generally for compact metric spaces $X$ that have an open subset, of full Hausdorff measure, with the structure of a Riemannian Lipschitz manifold. The basic analytic property of $\triangle_{*}$ that one would like to show is that $(I+\triangle_{*})^{-1}$ is compact; this implies discreteness of the spectrum of $\triangle_{*}$. In order to show that $(I+\triangle_{*})^{-1}$ is compact, it is necessary to make an additional assumption about $X$. To motivate this assumption, we recall that in a finite dimensional Alexandrov space $X$, any $x\in X$ has a neighborhood that is homeomorphic to the truncated tangent cone $T^{1}_{x}X$ [18, 27]. It seems likely that any $x\in X$ has a neighborhood that is biLipschitz homeomorphic to $T^{1}_{x}X$; this has been claimed, although no proof is available. Based on this, we consider a class ${\mathcal{C}}_{*}$ of compact metric spaces that are Lipschitz analogs of the topological multiconical spaces (MCS) introduced in [32] and used in [18, 27]. First, ${\mathcal{C}}_{0}$ consists of finite metric spaces. Inductively, if $X\in{\mathcal{C}}_{n}$ with $n\geq 1$ then any point in $X$ has a neighborhood that is biLipschitz homeomorphic to the truncated open metric cone over some element of ${\mathcal{C}}_{n-1}$ with diameter at most $\pi$. Conjecturally, any $n$-dimensional compact Alexandrov space is an element of ${\mathcal{C}}_{n}$. (If one is just interested in Alexandrov spaces then one can just start with elements of ${\mathcal{C}}_{0}$ consisting of one or two points. For boundaryless Alexandrov spaces, one can just start with elements of ${\mathcal{C}}_{0}$ consisting of two points.) Examples of elements of ${\mathcal{C}}_{*}$ come from quotients of smooth closed Riemannian manifolds by compact groups of isometries. Other examples come from compact stratified spaces with iterated cone-edge Riemannian metrics.

Theorem 1.16**.**

*(1) If $X\in{\mathcal{C}}_{n}$ then $\operatorname{Ker}(\triangle_{*})$ is isomorphic to $\operatorname{IH}^{GM}_{n-*}(X;{\mathcal{O}})$, the Goresky-MacPherson intersection homology of $X$ as defined using the upper middle perversity.

(2) If $X\in{\mathcal{C}}_{n}$ then $(I+\triangle_{*})^{-1}$ is compact.*

Here ${\mathcal{O}}$ is the orientation line bundle of the codimension-zero stratum of $X$. The upper middle perversity is the function $\overline{p}:{\mathbb{Z}}^{\geq 0}\rightarrow{\mathbb{Z}}$ given by $\overline{p}(0)=0$ and $\overline{p}(j)=\left[\frac{j-1}{2}\right]$ for $j\geq 1$. If $X$ is a conically stratified pseudomanifold (i.e. has no codimension-one strata) then it is well known that the $L^{2}$-cohomology of $X$ is related to the intersection (co)homology of $X$ with middle perversity. If $X$ is allowed to have codimension-one strata, as in our case, then there are various notions of intersection (co)homology [11]. It is not immediately clear which one is the right one to describe $\operatorname{Ker}(\triangle_{*})$. It turns out that the right one is the original Goresky-MacPherson intersection homology, extended to spaces with codimension-one strata, after an appropriate change of degree.

Remark 1.17*.*

A finite dimensional Alexandrov space is locally Lipschitz contractible [24]. Unfortunately, this does not help in proving Theorem 1.16 for Alexandrov spaces that are not a priori in ${\mathcal{C}}_{*}$, due to boundedness issues.

To summarize, we construct self-adjoint differential form Laplacians for a class of compact metric spaces, that includes compact Alexandrov spaces. For a more restricted class of compact metric spaces, that conjecturally includes compact Alexandrov spaces, we show that the differential form Laplacian has a compact resolvent.

The structure of the paper is as follows. In Section 2 we construct the differential form Laplacian $\triangle_{*}$ on a class of compact metric spaces. Section 3 has the construction of a sheaf of certain locally-$L^{2}$ differential forms. The eigenvalue bounds of Theorems 1.3 and 1.6 are proven in Section 4. In Section 5 we consider the Lipschitz multiconical spaces ${\mathcal{C}}_{*}$ and prove Theorem 1.16.

I thank Vitali Kapovitch for consultations on Alexandrov spaces, and Greg Friedman for consultations on intersection homology. I thank Bruno Colbois for a correction to an earlier version of the paper, and the referee for helpful remarks.

2. Differential form Laplacian on an Alexandrov space

In this section we define differential form Laplacians on a class of metric spaces that includes Alexandrov spaces. In Subsection 2.1 we consider a certain class of test forms built out of Lipschitz functions. Using them, in Subsection 2.2 we define a complex of $L^{2}$-forms. Subsection 2.3 has the construction of the differential form Laplacian.

For background material on Alexandrov spaces, we refer to [2, Chapter 10].

Let $(X,d_{X})$ be a compact metric space with Hausdorff dimension $n$ and finite $n$-dimensional Hausdorff mass. If $X$ is disconnected then we assume that the distance between points in distinct connected components is infinity. Suppose that there is an open subset $X^{*}\subset X$, with full Hausdorff $n$-measure, having the structure of an $n$-dimensional Riemannian Lipschitz manifold. This means that $X^{*}$ has a manifold structure with locally Lipschitz transition maps, and that it is equipped with a Riemannian metric $g$ so that in coordinate charts, $g$ and $g^{-1}$ are in $L^{\infty}_{loc}$. In addition, $d_{X}$ is compatible with the metric $d_{X^{*}}$ on $X^{*}$ coming from $g$ [8], in the sense that $d_{X}$ and $d_{X^{*}}$ coincide on some neighborhood of the diagonal in $X^{*}\times X^{*}$. In particular, if $F$ is a function with compact support in a coordinate neighborhood of $X^{*}$, and $F$ is Lipschitz in terms of the coordinates, then $F$ is a Lipschitz function on $X$.

Example 2.1*.*

Let $X$ be a compact Alexandrov space with curvature bounded below, of Hausdorff dimension $n$. There is some $\delta_{0}>0$ with the following property. Given $\delta\in(0,\delta_{0})$, let $X^{*}_{\delta}$ be the set of points $x\in X$ such that the space of directions $\Sigma_{x}$ has $(n-1)$-dimensional Hausdorff mass more than $(1-\delta)$ times that of $S^{n-1}$. Then $X^{*}_{\delta}$ is an open convex subset of $X$ of full Hausdorff measure, with the structure of a Riemannian Lipschitz manifold [26]. In fact, there is a stronger DC-structure, but this doesn’t seem to matter for the considerations of this paper.

2.1. Test forms

For a smooth compact Riemannian manifold, we can define an operator $d$, on a dense subset of $L^{2}$-forms, by saying that $\omega\in\operatorname{Dom}(d)$ if the distributional differential $d\omega$ is $L^{2}$. Here the notion of distributional differential uses smooth test forms. On our space $X$, it doesn’t make sense to talk about smooth forms. We will instead use “test forms” made from Lipschitz functions. (Not to be confused with the test forms mentioned in the introduction.) Let $\Omega^{*}_{\operatorname{Lip}}(X)$ be the graded-commutative differential graded algebra generated by $\{f_{0}df_{1}\ldots df_{k}\}$, where $f_{i}\in\operatorname{Lip}(X)$. In particular, an element of $\Omega^{k}_{\operatorname{Lip}}(X)$ is a finite sum of expressions $f_{0}df_{1}\ldots df_{k}$, and $d(f_{0}df_{1}\ldots df_{k})=1\cdot df_{0}df_{1}\ldots df_{k}$. There is a relation $d(fg)=fdg+gdf$ for $f,g\in\operatorname{Lip}(X)$. The elements of $\Omega^{*}_{\operatorname{Lip}}(X)$ are also known as the Kähler forms of the algebra $\operatorname{Lip}(X)$. There is a homomorphism $\rho$ from $\Omega^{*}_{\operatorname{Lip}}(X)$ to the locally-$L^{\infty}$ differential forms on $X^{*}$. This homomorphism need not be injective or surjective.

The test forms, or more precisely their image under $\rho$, will actually be twisted by the flat orientation line bundle ${\mathcal{O}}$ of $X^{*}$. The fiber of ${\mathcal{O}}$ over $x\in X^{*}$ is $\operatorname{H}^{n}(X^{*},X^{*}-x;{\mathbb{R}})$. If $X^{*}$ is orientable then with a given orientation $c$, the homomorphism $\rho_{c}$ to the ${\mathcal{O}}$-valued differential forms on $X^{*}$ can be identified with the $\rho$ of before. If $c^{\prime}$ is a different orientation then in the applications, $\rho_{c^{\prime}}(\omega^{\prime})$ will be equivalent to the result of changing $\rho_{c}(\omega^{\prime})$ by a sign on the components of $X^{*}$ where $c^{\prime}$ differs from $c$. We write $\Omega^{*}_{\operatorname{Lip}}(X;{\mathcal{O}})$ for the elements of $\Omega^{*}_{\operatorname{Lip}}(X)$ when we consider them to be twisted by ${\mathcal{O}}$.

If $X^{*}$ is not orientable then we only consider the case when $X$ is a boundaryless Alexandrov space. There is a notion of an orientation cover $\mathaccent 866{X}$ of $X$ [16]. It is also an Alexandrov space and is equipped with a ${\mathbb{Z}}_{2}$-action whose quotient is $X$. Choose an orientation on the connected space $\mathaccent 866{X}^{*}$. Then $\Omega^{*}_{\operatorname{Lip}}(X;{\mathcal{O}})\cong\Omega^{*}_{\operatorname{Lip}}(\mathaccent 866{X})\otimes_{{\mathbb{Z}}_{2}}{\mathbb{R}}$, where ${\mathbb{R}}$ has the nontrivial representation of ${\mathbb{Z}}_{2}$. (The papers [16, 23] discuss various equivalent notions of orientability for Alexandrov spaces.)

In the rest of the paper we will only discuss the case when $X^{*}$ is oriented, as the nonorientable case can be handled by working ${\mathbb{Z}}_{2}$-equivariantly on $\mathaccent 866{X}$.

Lemma 2.2**.**

If $\omega^{\prime}\in\Omega^{n-1}_{\operatorname{Lip}}(X;{\mathcal{O}})$ is such that $\rho(\omega^{\prime})$ and $\rho(d\omega^{\prime})$ have compact support in $X^{*}$ then $\int_{X^{*}}\rho(d\omega^{\prime})=0$.

Proof.

Put $K=\operatorname{supp}(\rho(\omega^{\prime}))\cup\operatorname{supp}(\rho(d\omega^{\prime}))$. Let $\{U_{i}\}_{i=1}^{N}$ be relatively compact coordinate neighborhoods of $X^{*}$ that cover K. Let $\{\phi_{i}\}_{i=1}^{N}$ be nonnegative subordinate Lipschitz functions whose sum is one on $K$. Write $\omega^{\prime}$ as a finite sum $\sum_{j}f_{0}^{j}df_{1}^{j}\ldots df_{n-1}^{j}$. Then

[TABLE]

and

[TABLE]

Hence it suffices to prove the lemma with $f_{0}^{j}$ replaced by $\phi_{i}f_{0}^{j}$, for some fixed $i$. Choose a Lipschitz function $\eta_{i}$ with compact support in $U_{i}$ so that $\eta_{i}\phi_{i}=\phi_{i}$, i.e. $\eta_{i}$ is one on $\operatorname{supp}(\phi_{i})$. Then

[TABLE]

and

[TABLE]

Hence we can reduce the lemma to the case when each of $f_{0}^{j},f_{1}^{j},\ldots,f_{n-1}^{j}$ has compact support in $U_{i}$. Using Euclidean coordinates on $U_{i}$, we can mollify the functions by convolution and take the mollification parameter to zero, to reduce to the case when $f_{0}^{j},f_{1}^{j},\ldots,f_{n-1}^{j}$ are smooth functions of the coordinates, in which case the lemma is evident. ∎

It is not immediately clear that $\rho(d\omega^{\prime})$ is determined by $\rho(\omega^{\prime})$, but this turns out to be the case.

Lemma 2.7**.**

Given $\omega^{\prime}_{1},\omega^{\prime}_{2}\in\Omega^{p}_{\operatorname{Lip}}(X)$, if $\rho(\omega^{\prime}_{1})=\rho(\omega^{\prime}_{2})$ then $\rho(d\omega^{\prime}_{1})=\rho(d\omega^{\prime}_{2})$

Proof.

It is equivalent to show that if $\rho(\omega^{\prime})=0$ then $\rho(d\omega^{\prime})=0$. Suppose that $\rho(\omega^{\prime})=0$. For any $\omega\in\Omega^{n-p-1}_{\operatorname{Lip}}(X;{\mathcal{O}})$ such that $\rho(\omega)$ and $\rho(d\omega)$ have compact support in $X^{*}$, Lemma 2.2 implies that

[TABLE]

Let $U$ be a relatively compact coordinate neighborhood for $X^{*}$. Let $F$ be a Lipschitz function with support in $U$. Let $\phi$ be a Lipschitz function with support in $U$ that is identically one on $\operatorname{supp}(F)$. Put $\omega=Fd(\phi x^{i_{1}})\ldots d(\phi x^{i_{p}})$. Then $\rho(\omega)=Fdx^{i_{1}}\wedge\ldots\wedge dx^{i_{p}}$. Letting $\omega$ vary over such choices, the lemma follows. ∎

Lemma 2.9**.**

If $\omega^{\prime}\in\Omega^{p}_{\operatorname{Lip}}(X)$ then $\operatorname{supp}(\rho(d\omega^{\prime}))\subset\operatorname{supp}(\rho(\omega^{\prime}))$.

Proof.

Suppose that $\operatorname{supp}(\rho(d\omega^{\prime}))$ is not contained in $\operatorname{supp}(\rho(\omega^{\prime}))$. Let $\omega\in\Omega^{n-p-1}(X;{\mathcal{O}})$ be such that $\rho(\omega)$ and $\rho(d\omega)$ have support in a relatively compact coordinate neighborhood of $X^{*}-\operatorname{supp}(\rho(\omega^{\prime}))$; such $\omega$ can be constructed as in the proof of Lemma 2.7. Then

[TABLE]

Letting $\omega$ vary over such choices gives a contradiction. ∎

For brevity, we will write $\omega^{\prime}$ for $\rho(\omega^{\prime})$ on $X^{*}$, and $d\omega^{\prime}$ for $\rho(d\omega^{\prime})$ on $X^{*}$. In what follows, this should not cause confusion.

2.2. $L^{2}$-complex

Let $\Omega^{*}_{L^{2}}(X)$ be the $L^{2}$-differential forms on $X^{*}$. There is a well-defined integration $\int_{X*}:\Omega^{n}_{L^{2}}\left(X;{\mathcal{O}}\right)\rightarrow{\mathbb{R}}$.

The map $\rho$ sends $\Omega^{*}_{\operatorname{Lip}}(X)$ to $\Omega^{*}_{L^{2}}(X)$.

Lemma 2.11**.**

The image of $\rho$ is dense in $\Omega^{*}_{L^{2}}(X)$.

Proof.

As in the proof of Lemma 2.7, let $U$ be a relatively compact coordinate neighborhood and let $F$ be a Lipschitz function with support in $U$. Then $Fdx^{i_{1}}\wedge\ldots\wedge dx^{i_{p}}$ is in the image of $\rho$, from which the lemma follows. ∎

Let $\Omega^{p}_{L^{2},d}(X)$ be the elements $\omega\in\Omega^{p}_{L^{2}}(X)$ for which there is some $\eta\in\Omega^{p+1}_{L^{2}}(X)$ so that for all $\omega^{\prime}\in\Omega^{n-p-1}_{\operatorname{Lip}}(X;{\mathcal{O}})$, we have

[TABLE]

If such an $\eta$ exists then it is unique, and we put $d\omega=\eta$. This defines a map $d:\Omega^{p}_{L^{2},d}(X)\rightarrow\Omega^{p+1}_{L^{2}}(X)$. (In the case when $X$ is a smooth closed Riemannian manifold, the definition of $d$ is similar to how one defines the maximal closed extension of the exterior derivative on smooth forms.)

Example 2.13*.*

If $X=[0,1]$ then $\Omega^{0}_{L^{2},d}([0,1])=\{f\in H^{1}([0,1]):f(0)=f(1)=0\}$ and $\Omega^{1}_{L^{2},d}([0,1])=\Omega^{1}_{L^{2}}([0,1])$. More generally, if $X$ is a smooth compact Riemannian manifold-with-boundary, with boundary inclusion $i:\partial X\rightarrow X$, then an element $\omega$ of $\Omega^{*}_{L^{2},d}(X)$ has a well-defined restriction $i^{*}\omega$ in $\Omega^{*}(\partial X)/\overline{\operatorname{Im}(d)}$ that vanishes. These are relative (Dirichlet) boundary conditions.

Remark 2.14*.*

If we replace $X^{*}$ by an open subset of $X^{*}$ with full measure then $\Omega^{p}_{L^{2}}(X)$ and $\Omega^{p}_{L^{2},d}(X)$ do not change. (This would not be the case if we required $\omega^{\prime}$ to have support in a compact subset of $X^{*}$.) As a consequence, $\Omega^{p}_{L^{2}}(X)$ and $\Omega^{p}_{L^{2},d}(X)$ are independent of the choice of $X^{*}$. Namely, if $X^{*}_{1}$ and $X^{*}_{2}$ are two different choices then in each case, the ensuing spaces $\Omega^{p}_{L^{2}}(X)$ and $\Omega^{p}_{L^{2},d}(X)$ are the same as those coming from $X^{*}_{3}=X^{*}_{1}\cap X^{*}_{2}$.

In particular, if $X$ is a compact $n$-dimensional Alexandrov space, let $X^{*}_{\delta}$ be the subspace of Example 2.1. If $\delta^{\prime}<\delta$ then $X^{*}_{\delta^{\prime}}$ is an open subset of $X^{*}_{\delta}$ with full measure. Hence the notions of $\Omega^{p}_{L^{2}}(X)$ and $\Omega^{p}_{L^{2},d}(X)$ are independent of $\delta$.

Lemma 2.15**.**

The subspace $\Omega^{p}_{L^{2},d}(X)$ is dense in $\Omega^{p}_{L^{2}}(X)$.

Proof.

It suffices to prove the corresponding statement for the elements of $\Omega^{p}_{L^{2}}(X)$ with support in a fixed but arbitrary compact set $K\subset X^{*}$. If $\{\sigma_{i}\}_{i=1}^{N}$ are Lipschitz functions on $X$ and $\{\omega_{i}\}_{i=1}^{N}$ are elements of $\Omega^{p}_{L^{2},d}(X)$ then one can check that $\sum_{i=1}^{N}\sigma_{i}\omega_{i}\in\Omega^{p}_{L^{2},d}(X)$, with $d\sum_{i=1}^{N}\sigma_{i}\omega_{i}=\sum_{i=1}^{N}(d\sigma_{i}\wedge\omega_{i}+\sigma_{i}d\omega_{i})$. Using a covering of $K$ by relatively compact coordinate neighborhoods of $X^{*}$, and nonnegative subordinate Lipschitz functions whose sum is one on $K$, we can reduce to the case when $K$ is a closed ball in a fixed coordinate neighborhood. Considering forms with support in $K$ that are smooth with respect to the given coordinates, the lemma follows. ∎

Lemma 2.16**.**

The operator $d:\Omega^{p}_{L^{2},d}(X)\rightarrow\Omega^{p+1}_{L^{2}}(X)$ is closed.

Proof.

Suppose that $\{\omega_{i}\}_{i=1}^{\infty}$ is a sequence in $\Omega^{p}_{L^{2},d}(X)$ so that there is a limit of pairs $\lim_{i\rightarrow\infty}(\omega_{i},d\omega_{i})=(\omega_{\infty},\eta_{\infty})$ for some $(\omega_{\infty},\eta_{\infty})\in\Omega^{p}_{L^{2}}(X)\oplus\Omega^{p+1}_{L^{2}}(X)$. Replacing $\omega$ in (2.12) by $\omega_{i}$ and passing to the limit shows that $\omega_{\infty}\in\Omega^{p}_{L^{2},d}(X)$ and $\eta_{\infty}=d\omega_{\infty}$. This proves the lemma. ∎

Lemma 2.17**.**

The image of $d:\Omega^{p}_{L^{2},d}(X)\rightarrow\Omega^{p+1}_{L^{2}}(X)$ lies in $\Omega^{p+1}_{L^{2},d}(X)$, and $\overline{\operatorname{Im}(d)}\subset\operatorname{Ker}(d)$.

Proof.

Given $\omega\in\Omega^{p}_{L^{2},d}(X)$, replacing $\omega^{\prime}$ in (2.12) with $d\omega^{\prime}$ gives

[TABLE]

for all $\omega^{\prime}\in\Omega_{\operatorname{Lip}}^{n-p-2}(X;{\mathcal{O}})$. It follows that $\operatorname{Im}(d)\subset\Omega^{p+1}_{L^{2},d}(X)$ and $d^{2}=0$. Since $d$ is a closed operator, $\operatorname{Ker}(d)$ is a closed subset of $\Omega^{p+1}_{L^{2}}(X)$. Hence $\overline{\operatorname{Im}(d)}\subset\operatorname{Ker}(d)$. ∎

2.3. Differential form Laplacian

Define a quadratic form on $\Omega^{p}_{L^{2},d}(X)/\overline{\operatorname{Im}(d)}\subset\Omega^{p}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$ by

[TABLE]

As $d$ is closed, it follows that $Q$ is a closed quadratic form in the sense of [28, Section VIII.6]. There is a corresponding self-adjoint operator $\triangle_{p}=d^{*}d$, densely defined on $\Omega^{p}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$ [28, Theorem VIII.15], [29, Theorem X.25]. Its domain is

[TABLE]

We have isometric isomorphisms

[TABLE]

and

[TABLE]

Using the isomorphisms

[TABLE]

and

[TABLE]

we obtain an orthogonal decomposition

[TABLE]

We have already defined $\triangle_{p}$ on

[TABLE]

We can define the Laplacian on $\overline{\operatorname{Im}(d)}\subset\Omega^{*}_{L^{2}}(X)$ by using the isomorphism $d:\Omega^{*-1}_{L^{2}}(X)/\operatorname{Ker}(d)\rightarrow\operatorname{Im}(d)$ to transfer the Laplacian from $\Omega^{*-1}_{L^{2}}(X)/\operatorname{Ker}(d)$. In this way, there is a Laplacian $\triangle^{Hodge}_{*}$ on $\Omega^{*}_{L^{2}}(X)$ with a full Hodge decomposition. Using this Hodge decomposition, spectral questions for $\triangle^{Hodge}_{*}$ reduce to spectral questions about the Laplacian $\triangle_{*}$ on $\Omega^{*}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$. In the rest of the paper, we mostly concentrate on the latter Laplacian.

Example 2.27*.*

If $X$ is a compact Riemannian manifold-with-boundary then $\triangle_{p}$ is the densely-defined Laplacian on the Hilbert space $\Omega^{p}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$, with relative (Dirichlet) boundary conditions, e.g. $\operatorname{Ker}(\triangle_{p})\cong\operatorname{H}^{p}(X,\partial X;{\mathbb{R}})$.

Example 2.28*.*

We give an example in which $\triangle_{p}$ has an infinite dimensional kernel. Start with the cone $(0,1]\times S^{3}$, equipped with the metric $g=dr^{2}+r^{2}g_{S^{3}}$. Glue a $4$-ball onto the $S^{3}$-boundary. Choose some point $m\in S^{3}$ and for each $i>1$, perform a connected sum with a copy of ${\mathbb{C}}P^{2}$, with size comparable to $100^{-i}$, at the point $(i^{-1},m)$ in the conical region. Call the result $X^{*}$ and let $X$ be its $1$-point compactification. Now $\operatorname{Im}\left(\operatorname{H}^{p}_{c}(X^{*})\rightarrow\operatorname{H}^{p}(X^{*})\right)$ injects into $\operatorname{Ker}(d:\Omega^{p}_{L^{2},d}(X)\rightarrow\Omega^{p+1}_{L^{2},d}(X))/\overline{\operatorname{Im}(d:\Omega^{p-1}_{L^{2},d}(X)\rightarrow\Omega^{p}_{L^{2},d}(X))}$; see [22, Proposition 4], whose proof does not need completeness of $X^{*}$. As $\operatorname{Im}\left(\operatorname{H}^{2}_{c}(X^{*})\rightarrow\operatorname{H}^{2}(X^{*})\right)$ is infinite dimensional, it follows that $\operatorname{Ker}(\triangle_{2})$ is infinite dimensional.

Remark 2.29*.*

An alternative differential form Laplacian can be defined using the closure of the differential on $\Omega^{*}_{\operatorname{Lip}}(X)$ (sometimes called the minimal closed extension). Namely, say that an element $\omega\in\Omega^{*}_{L^{2}}(X)$ lies in $\operatorname{Dom}(d)$ if there is a sequence $\omega_{i}\in\Omega^{*}_{\operatorname{Lip}}(X)$ such that $\lim_{i\rightarrow\infty}\omega_{i}=\omega$ in $\Omega^{*}_{L^{2}}(X)$, and $\lim_{i\rightarrow\infty}d\omega_{i}$ exists in $\Omega^{*+1}_{L^{2}}(X)$. If this is the case, put $d\omega=\lim_{i\rightarrow\infty}d\omega_{i}$; it is independent of the particular choice of $\{\omega_{i}\}_{i=1}^{\infty}$. Then $d$ is a closed operator and one can consider $d^{*}d$, acting on $\Omega^{*}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$.

If $X$ is a compact Riemannian manifold-with-boundary then one recovers the differential form Laplacian on $\Omega^{*}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$, with absolute (Neumann) boundary conditions, this way. The differential form Laplacian $\triangle_{*}$, as defined following (2.19), is more convenient for the purposes of this paper, as will be seen in the proof of Theorem 1.6.

Remark 2.30*.*

If $X$ is a smooth closed Riemannian manifold then there is a Hodge Laplacian $\triangle^{Hodge}_{p}=dd^{*}+d^{*}d$ acting on the $H^{2}$-regular $p$-forms. The space of $H^{2}$-regular $p$-forms is independent of the particular Riemannian metric.

If $X$ is a closed Riemannian Lipschitz manifold then a Hodge Laplacian $\triangle^{Hodge}_{*}$, with dense domain in $\Omega^{*}_{L^{2}}(X)$, was defined in [33]. However, there are some subtleties. For example, the corresponding quadratic form

[TABLE]

has domain $\operatorname{Dom}(Q^{Hodge})=\{\omega\in\Omega^{*}_{L^{2}}(X)\>:\>d\omega\in\Omega^{*+1}_{L^{2}}(X),d*\omega\in\Omega^{n-*+1}_{L^{2}}(X)\}$. Due to the appearance of the Hodge duality operator $*$ in $d*\omega$, the domain of $Q^{Hodge}$ definitely depends on the precise $L^{\infty}_{loc}$-Riemannian metric used on $X$ [33, p. 46].

In contrast, the quadratic form $Q$ of (2.19) has domain $\Omega^{*}_{L^{2},d}(X)/\overline{\operatorname{Im}(d)}$ which, in the case of a closed Riemannian Lipschitz manifold, is independent of the precise Riemannian metric. Hence the domain of $\triangle_{*}$ is also independent of the precise Riemannian metric. This is one manifestation of the fact that $\triangle_{*}$ has better biLipschitz properties than $\triangle^{Hodge}_{*}$.

3. $L^{2}$-sheaf

In this section we define a sheaf $\Omega^{*}_{L^{2}_{loc},d}$ of differential graded complexes, constructed from certain locally-$L^{2}$ differential forms. The result of this section will be used in Section 5.

We continue with the setup of Section 2. In particular, $X$ is a compact metric space and $X^{*}\subset X$ is an open subset of full Hausdorff measure, with the structure of a Riemannian Lipschitz manifold.

Given an open set $U\subset X$, let $\Omega^{*}_{L^{2}_{loc}}(U)$ be the locally-$L^{2}$ differential forms on $U^{*}=X^{*}\cap U$. Let $\Omega^{p}_{L^{2}_{loc},d}(U)$ be the elements $\omega\in\Omega^{p}_{L^{2}_{loc}}(U)$ for which there is some $\eta\in\Omega^{p+1}_{L^{2}_{loc}}(U)$ so that for all compact subsets $K\subset U$ and all $\omega^{\prime}\in\Omega^{n-p-1}_{\operatorname{Lip}}(X;{\mathcal{O}})$ with support in $K$, we have

[TABLE]

If such an $\eta$ exists then it is unique, and we put $d\omega=\eta$. Note that an element of $\Omega^{p}_{L^{2}_{loc},d}(U)$ may not satisfy relative boundary conditions in any sense.

Lemma 3.2**.**

The assignment $U\rightarrow\Omega^{p}_{L^{2}_{loc},d}(U)$ defines a sheaf $\Omega^{p}_{L^{2}_{loc},d}$ on $X$.

Proof.

Given $V\subset U$, there is clearly a restriction map $r_{V,U}:\Omega^{p}_{L^{2}_{loc},d}(U)\rightarrow\Omega^{p}_{L^{2}_{loc},d}(V)$ that defines a presheaf. Let $\{U_{\alpha}\}$ be an open covering of $U$. Given elements $\omega_{1},\omega_{2}\in\Omega^{p}_{L^{2}_{loc},d}(U)$, if $r_{U_{\alpha},U}(\omega_{1})=r_{U_{\alpha},U}(\omega_{2})$ for all $\alpha$, then $\omega_{1}=\omega_{2}$.

Now suppose that $\omega_{\alpha}\in\Omega^{p}_{L^{2}_{loc},d}(U_{\alpha})$ are such that $r_{U_{\alpha}\cap U_{\beta},U_{\alpha}}(\omega_{\alpha})=r_{U_{\alpha}\cap U_{\beta},U_{\beta}}(\omega_{\beta})$ for all $\alpha$ and $\beta$. There is a unique $\omega\in\Omega^{p}_{L^{2}_{loc}}(U)$ so that $\omega_{\alpha}=r_{U_{\alpha},U}(\omega)$ for all $\alpha$. We have to show that $\omega\in\Omega^{p}_{L^{2}_{loc},d}(U)$. Let $K$ be a compact subset of $U$. Then $K$ is covered by a finite subset $\{U_{i}\}_{i=1}^{N}$ of the $U_{\alpha}$’s. Let $\{\sigma_{i}\}_{i=1}^{N}$ be nonnegative subordinate Lipschitz functions whose sum is one on $K$. In particular, the support of $\sigma_{i}$ is a compact set $K_{i}\subset U_{i}$. For any $\omega^{\prime}\in\Omega^{n-p-1}_{\operatorname{Lip}}(X;{\mathcal{O}})$ with support in $K$, since $\sigma_{i}\omega^{\prime}\in\Omega^{n-p-1}_{\operatorname{Lip}}(X;{\mathcal{O}})$ has support in $U_{i}$, and $\omega_{i}\in\Omega^{p}_{L^{2}_{loc},d}(U_{i})$, we have

[TABLE]

Summing over $i$ gives

[TABLE]

Thus $\omega\in\Omega^{p}_{L^{2}_{loc},d}(U)$, with $d\omega=\sum_{i=1}^{N}\sigma_{i}d\omega_{i}$. This proves the lemma. ∎

4. Eigenvalue estimates

In this section we prove the eigenvalue estimates of Theorems 1.3 and 1.6. In both cases, the proof is by a contradiction argument.

As mentioned in Remark 1.11, Theorem 1.3 actually follows from Theorem 1.6. As the proof of Theorem 1.3 makes the strategy clearer, we give it first, in Subsection 4.1. To get the upper eigenvalue bound, we pullback forms from a nice subset of a limiting Alexandrov space that has the same dimension $n$ as the manifolds. In Subsection 4.2 we prove Theorem 1.6. The proof is more involved, in that we need to pullback forms from an Alexandrov space that may be of a lower dimension. We show that there is a region in the limiting Alexandrov space that is almost Euclidean, and above which the approximating manifold has a fibration structure with controlled geometry. We then pullback forms as in the proof of Theorem 1.3.

4.1. Noncollapsing case

Let $X$ be a compact metric space of the type considered in Section 2. Given $k\in{\mathbb{Z}}^{+}$ and $p\in[0,n]$, put

[TABLE]

where $V_{k,p}$ ranges over $k$-dimensional subspaces of $\Omega^{p}_{L^{2},d}(X)/\operatorname{Ker}(d)$. If $\triangle_{p}$ has discrete spectrum with finite multiplicities then $\lambda_{k,p}(X)$ is the $k^{th}$ eigenvalue of $\triangle_{p}$ on $\Omega^{p}_{L^{2}}(X)/\operatorname{Ker}(d)$, counted with multiplicity [30, Theorem XIII.2].

Lemma 4.2**.**

If $X_{1}$ and $X_{2}$ are $C$-biLipschitz then

[TABLE]

Proof.

By reducing $X_{1}^{*}$ and $X_{2}^{*}$ if necessary, we can assume that the biLipschitz map $h:X_{1}\rightarrow X_{2}$ restricts to a biLipschitz map between $X_{1}^{*}$ and $X_{2}^{*}$. Then on $X_{1}^{*}$, we have $C^{-2}g_{1}\leq h^{*}g_{2}\leq C^{2}g_{1}$. There is a bounded pullback map $h^{*}:\Omega^{*}_{L^{2}}(X_{2})\rightarrow\Omega^{*}_{L^{2}}(X_{1})$. One can check that $h^{*}$ sends $\Omega^{*}_{L^{2},d}(X_{2})$ to $\Omega^{*}_{L^{2},d}(X_{1})$, with $h^{*}d=dh^{*}$. Hence there is also a pullback $h^{*}:\Omega^{p}_{L^{2},d}(X_{2})/\operatorname{Ker}(d_{X_{2}})\rightarrow\Omega^{p}_{L^{2},d}(X_{1})/\operatorname{Ker}(d_{X_{1}})$.

Given a $k$-dimensional subspace $V_{k,p}$ of $\Omega^{p}_{L^{2},d}(X_{2})/\operatorname{Ker}(d_{X_{2}})$ and a nonzero element $\mathaccent 869{\psi}\in V_{k,p}$, let ${\psi}\in\Omega^{p}_{L^{2},d}(X_{2})$ be a representative for $\mathaccent 869{\psi}$. Then

[TABLE]

and

[TABLE]

It follows that

[TABLE]

Reversing the roles of $X_{1}$ and $X_{2}$ gives (4.3). This proves the lemma. ∎

We now prove a version of Theorem 1.3 for compact Alexandrov spaces.

Proposition 4.7**.**

Given $n\in{\mathbb{Z}}^{+}$, $K\in{\mathbb{R}}$ and $v>0$, there is some $L=L(n,K,v)<\infty$ so that for any $n$-dimensional compact connected Alexandrov space $X$ for which

1. (1)

the curvature of $X$ is bounded below by $K$, 2. (2)

$\operatorname{diam}(X)\leq 1$* and* 3. (3)

$\operatorname{vol}(X)\geq v$,

and any $p\in[0,n-1]$ and $k\in{\mathbb{Z}}^{+}$, we have $\lambda_{k,p}(X)\leq Lk^{\frac{2}{n}}$. Here $\lambda_{k,p}(X)$ is defined by (4.1).

Proof.

Suppose that the claim about $\lambda_{k,p}$ is not true. Then there is a sequence $\{X_{i}\}_{i=1}^{\infty}$ of $n$-dimensional Alexandrov spaces and some $p\in[0,n-1]$ so that

1. (1)

the curvature of $X_{i}$ is bounded below by $K$, 2. (2)

$\operatorname{diam}(X_{i})\leq 1$ and 3. (3)

$\operatorname{vol}(X_{i})\geq v$, but 4. (4)

$\lambda_{k_{i},p}(X_{i})\geq ik_{i}^{\frac{2}{n}}$ for some $k_{i}\in{\mathbb{Z}}^{+}$.

Let $c_{i}$ be the smallest integer such that $k_{i}\leq 2^{nc_{i}}$. Then

[TABLE]

Putting $\overline{k}_{i}=2^{nc_{i}}$, we have $\lambda_{\overline{k}_{i},p}(X_{i})\geq\frac{1}{4}i\overline{k}_{i}^{\frac{2}{n}}$.

After passing to a subsequence, we can assume that $\lim_{i\rightarrow\infty}X_{i}=X_{\infty}$ in the Gromov-Hausdorff topology, where $X_{\infty}$ is also an $n$-dimensional Alexandrov space. Let $x_{\infty}$ be a regular point of $X_{\infty}$. Then there is a neighborhood $U_{\infty}$ of $x_{\infty}$, that is biLipschitz homeomorphic to $(0,1)^{n}\subset{\mathbb{R}}^{n}$, such that for large $i$, there are

• •

points $x_{i}\in X_{i}$,

• •

neighborhoods $U_{i}$ of $x_{i}$, and

• •

bijective maps $\sigma_{i}:U_{i}\rightarrow U_{\infty}$

that are uniformly biLipschitz [3, Theorem 9.8].

Letting $h_{i}$ be the composition of $\sigma_{i}$ with the biLipschitz homeomorphism from $U_{\infty}$ to $(0,1)^{n}$, the maps $h_{i}:U_{i}\rightarrow(0,1)^{n}$ are $\Lambda$-biLipschitz for some $\Lambda<\infty$ independent of $i$.

Let $\psi$ be a smooth compactly supported $p$-form on $(0,1)^{n}$ with $\int_{(0,1)^{n}}|\psi|^{2}\>d^{n}x=1$ and $d\psi\neq 0$. Put $E_{\psi}=\int_{(0,1)^{n}}|d\psi|^{2}\>d^{n}x$.

Let $\operatorname{Ker}(d_{(0,1)^{n}})$ be the forms $\mu\in\Omega^{p}_{L^{2}}((0,1)^{n})$ such that

[TABLE]

for all compactly supported $\omega^{\prime}\in\Omega^{n-p-1}_{\operatorname{Lip}}((0,1)^{n})$. Then $\operatorname{Ker}(d_{(0,1)^{n}})$ is closed in $\Omega^{p}_{L^{2}}((0,1)^{n})$.

Lemma 4.10**.**

$\psi$* does not lie in $\operatorname{Ker}(d_{(0,1)^{n}})$.*

Proof.

By assumption, $d\psi\neq 0$ as a smooth form. Hence we can find some $\omega^{\prime}\in\Omega^{n-p-1}_{\operatorname{Lip}}((0,1)^{n})$ so that $\int_{(0,1)^{n}}d\omega^{\prime}\wedge\psi=(-1)^{n-p}\int_{(0,1)^{n}}\omega^{\prime}\wedge d\psi\neq 0$. ∎

With reference to Lemma 4.10, let $N_{\psi}$ be the square of the norm of the image of $\psi$ in $\Omega^{p}_{L^{2}}((0,1)^{n})/\operatorname{Ker}(d_{(0,1)^{n}})$.

Let $R_{i}:(0,2^{-c_{i}})^{n}\rightarrow(0,1)^{n}$ be multiplication by $2^{c_{i}}$. Under rescaling, $\int_{(0,2^{-c_{i}})^{n}}|dR_{i}^{*}\psi|^{2}\>dx^{n}=2^{c_{i}(2p+2-n)}E_{\psi}$ and the square norm of $R_{i}^{*}\psi$ in $\Omega^{p}_{L^{2}}((0,2^{-c_{i}})^{n})/\operatorname{Ker}(d)$ is $2^{c_{i}(2p-n)}N_{\psi}$.

There are $\overline{k}_{i}=2^{nc_{i}}$ disjoint boxes $\{B_{j}\}_{j=1}^{\overline{k}_{i}}$ in $(0,1)^{n}$, each congruent to $(0,2^{-c_{i}})^{n}$. Let $\psi_{j}$ be the translate of $R_{i}^{*}\psi$ to $B_{j}$. Let $V^{\infty}_{i}$ be the span of $\{\psi_{j}\}_{j=1}^{\overline{k}_{i}}$ in $\Omega^{p}_{L^{2}}((0,1)^{n})$, let ${h_{i}^{*}V^{\infty}_{i}}$ denote the extension by zero from $\Omega^{p}_{L^{2},d}(U_{i})$ to $\Omega^{p}_{L^{2},d}(X_{i})$ of the pullback, and let $\mathaccent 1373{h_{i}^{*}V^{\infty}_{i}}$ denote the image of $h_{i}^{*}V^{\infty}_{i}$ in $\Omega^{p}_{L^{2},d}(X_{i})/\operatorname{Ker}(d_{X_{i}})$. We claim that $\mathaccent 1373{h_{i}^{*}V^{\infty}_{i}}$ is $\overline{k}_{i}$-dimensional. To see this, if there is a relation $\sum_{j=1}^{\overline{k}_{i}}\alpha_{j}h_{i}^{*}\psi_{j}\in\operatorname{Ker}(d_{X_{i}})$ then

[TABLE]

and hence $\sum_{j=1}^{\overline{k}_{i}}\alpha_{j}d\psi_{j}=0$, which implies that each $\alpha_{j}$ vanishes.

If $\eta_{i}=\sum_{j=1}^{\overline{k}_{i}}\beta_{j}h_{i}^{*}\psi_{j}$ is a nonzero element of $h_{i}^{*}V^{\infty}_{i}$ then as in the proof of (4.4), for large $i$, we have

[TABLE]

Let $\mathaccent 869{\eta}_{i}$ be the class of $\eta_{i}$ in $\mathaccent 1373{h_{i}^{*}V^{\infty}_{i}}$. We claim that

[TABLE]

To see this, suppose that $\|\mathaccent 869{\eta}_{i}\|^{2}<\frac{1}{4}2^{c_{i}(2p-n)}N_{\psi}\Lambda^{-2p-n}\sum_{j=1}^{\overline{k}_{i}}|\beta_{j}|^{2}$. Then there is some $\sigma\in\operatorname{Ker}(d_{X_{i}})$ so that

[TABLE]

As in (4.5), it follows that

[TABLE]

However,

[TABLE]

This is a contradiction.

Combining (4.12) and (4.13) gives

[TABLE]

For large $i$, this contradicts the fact that $\lambda_{\overline{k}_{i},p}(X_{i})\geq\frac{1}{4}i\overline{k}_{i}^{\frac{2}{n}}$. This proves the proposition. ∎

Corollary 4.18**.**

Given $n\in{\mathbb{Z}}^{+}$, $K\in{\mathbb{R}}$ and $v>0$, there is some $L=L(n,K,v)<\infty$ with the following property. Let $M$ be an $n$-dimensional compact connected Riemannian manifold-with-boundary whose boundary, if nonempty, is convex. Suppose that

1. (1)

the sectional curvature of $M$ is bounded below by $K$, 2. (2)

$\operatorname{diam}(M)\leq 1$* and* 3. (3)

$\operatorname{vol}(M)\geq v$.

Then for any $p\in[0,n]$ and any $k\in{\mathbb{Z}}^{+}$, the $k^{th}$ positive eigenvalue of the $p$-form Laplacian on $M$, as defined with relative boundary conditions, satisfies $\lambda_{k,p}(M)\leq Lk^{\frac{2}{n}}$.

Proof.

Proposition 4.7 gives upper bounds on the eigenvalues of the Laplacian on $\Omega^{p}_{L^{2}}(M)/\operatorname{Ker}(d)$ for all $p\in[0,n-1]$. From the Hodge decomposition, we also get an upper bound on the eigenvalues of the Laplacian on $\operatorname{Im}(d)\subset\Omega^{n}_{L^{2}}(M)$. The corollary follows. ∎

Theorem 1.3 follows from Corollary 4.18.

4.2. General case

We now prove Theorem 1.6.

Proposition 4.19**.**

Given $n\in{\mathbb{Z}}^{+}$ and $K\in{\mathbb{R}}$, there is some $A=A(n,K)<\infty$ with the following property. Let $M$ be an $n$-dimensional closed connected Riemannian manifold for which

1. (1)

the sectional curvature of $M$ is bounded below by $K$ and 2. (2)

there is some point $m\in M$ with an $s$-strainer of quality $\frac{1}{10}$ and size $1$, where $1\leq s\leq n$.

Then for any $p\in[0,s-1]$ and $k\in{\mathbb{Z}}^{+}$, we have $\lambda_{k,p}(M)\leq Ak^{\frac{2}{s}}$. Here $\lambda_{k,p}(M)$ is the $k^{th}$ eigenvalue of the Laplacian on $\Omega^{p}_{L^{2}}(M)/\operatorname{Ker}(d)$.

Proof.

Suppose that the claim about $\lambda_{k,p}$ is not true. Then for some $n\in{\mathbb{Z}}^{+}$, some $s\in[1,n]$ and some $p\in[0,s-1]$, there is a sequence $\{M_{i}\}_{i=1}^{\infty}$ of $n$-dimensional closed Riemannian manifolds and numbers $k_{i}\in{\mathbb{Z}}^{+}$ so that

1. (1)

the curvature of $M_{i}$ is bounded below by $K$ and 2. (2)

there is some point $m_{i}\in M_{i}$ with an $s$-strainer of quality $\frac{1}{10}$ and size $1$, but 3. (3)

$\lambda_{k_{i},p}(M_{i})\geq ik_{i}^{\frac{2}{s}}$.

Let $c_{i}$ be the smallest integer such that $k_{i}\leq 2^{n!c_{i}}$. Then

[TABLE]

Putting $\overline{k}_{i}=2^{n!c_{i}}$, we have $\lambda_{\overline{k}_{i},p}(M_{i})\geq\frac{i}{4^{\frac{n!}{s}}}\overline{k}_{i}^{\frac{2}{s}}$.

After passing to a subsequence, we can assume that $\lim_{i\rightarrow\infty}(M_{i},m_{i})=(X_{\infty},x)$ in the pointed Gromov-Hausdorff topology, where $X_{\infty}$ is a complete Alexandrov space, say of dimension $n_{\infty}$. From the strainer condition, $n_{\infty}\geq s$. Let $x_{\infty}$ be a regular point of $X_{\infty}$, say within distance $\frac{1}{1000}$ from $x$.

Given $\delta>0$, we can find a $n_{\infty}$-strainer $\{a_{l},b_{l}\}_{l=1}^{n_{\infty}}$ of quality $\delta$ around $x_{\infty}$, say of size $r_{\delta}$, with $\lim_{\delta\rightarrow 0}r_{\delta}=0$.

I thank Vitali Kapovitch for the proof of the next lemma.

Lemma 4.21**.**

If $\delta$ is sufficiently small then for large $i$, the following holds. There are an open subset $U_{i}$ of $M_{i}$, a closed Lipschitz manifold $Z_{i}$, a Lipschitz surjection $\tau_{i}:U_{i}\rightarrow B(0,\delta r_{\delta})$ and a commutative diagram

[TABLE]

where $\alpha_{i}$ is a biLipschitz homeomorphism. Furthermore, $\tau_{i}$ is an almost metric submersion in the sense of [35, p. 318].

Proof.

Let $\mathaccent 869{m}_{i}\in M_{i}$ be such that $\{\mathaccent 869{m}_{i}\}_{l=1}^{n_{\infty}}$ converges to $x_{\infty}$. Given $\delta>0$, for large $i$, let $\{\mathaccent 869{a}_{i,l},\mathaccent 869{b}_{i,l}\}_{l=1}^{n_{\infty}}$ be an $n_{\infty}$-strainer of quality $2\delta$ in $M_{i}$ such that as $i\rightarrow\infty$, $\{\mathaccent 869{a}_{i,l}\}_{l=1}^{n_{\infty}}$ converges to $\{a_{l}\}_{l=1}^{n_{\infty}}$ and $\{\mathaccent 869{b}_{i,l}\}_{l=1}^{n_{\infty}}$ converges to $\{b_{l}\}_{l=1}^{n_{\infty}}$. Define ${\gamma}_{i}:B(\mathaccent 869{m}_{i},r_{\delta})\rightarrow{\mathbb{R}}^{n_{\infty}}$ by ${\gamma}_{i}(p_{i})=\{d(\mathaccent 869{m}_{i},\mathaccent 869{a}_{i,l})-d(p_{i},\mathaccent 869{a}_{i,l})\}_{l=1}^{n_{\infty}}$.

Consider the pointed Riemannian manifolds $\{(M_{i},r_{\delta}^{-2}g_{i},\mathaccent 869{m}_{i})\}_{i=1}^{\infty}$. The rescaled limit space $r_{\delta}^{-1}X_{\infty}$ has a strainer of size $1$ and quality $\delta$, centered at $x_{\infty}$. There is some $\rho>0$ so that as $\delta\rightarrow 0$, the $\rho$-balls around $x_{\infty}$ in $r_{\delta}^{-1}X_{\infty}$ converge in the pointed Gromov-Hausdorff topology to $B(0,\rho)\subset{\mathbb{R}}^{n_{\infty}}$. It follows that for any $\epsilon>0$, there is a $\delta_{0}>0$ so that if $\delta<\delta_{0}$ then for all large $i$, the map $r_{\delta}^{-1}\gamma_{i}:B(\mathaccent 869{m}_{i},r_{\delta})\rightarrow{\mathbb{R}}^{n_{\infty}}$ defines a pointed $\epsilon$-Gromov-Hausdorff approximation between $(\gamma_{i}^{-1}(B(0,\rho r_{\delta})),\mathaccent 869{m}_{i})$ and $(B(0,\rho),0)$, where $\gamma_{i}^{-1}(B(0,\rho r_{\delta}))$ has the restricted metric from $(M_{i},r_{\delta}^{-2}g_{i})$. For an appropriate choice of $\epsilon$ and taking $\delta$ small enough, we can apply [35] in the pointed setting to get the Lipschitz fibration, along with the almost metric submersion property. (In fact, the fibration in [35] is $C^{1}$.) As $B(0,\delta r_{\delta})$ is contractible, the fibration structure is a product structure. ∎

Since $\tau_{i}$ is Lipschitz and surjective, for almost all $u\in B(0,\delta r_{\delta})$ and almost all $u_{i}\in\tau_{i}^{-1}(u)$, the differential $d\tau_{i}:T_{u_{i}}M_{i}\rightarrow{\mathbb{R}}^{n_{\infty}}$ is defined. Given such a point $u_{i}\in U_{i}$, put $V_{u_{i}}=\operatorname{Ker}((d\tau_{i})_{u_{i}})$, an $(n-n_{\infty})$-dimensional subspace of $T_{u_{i}}U_{i}$. Put $H_{u_{i}}=V_{u_{i}}^{\perp}$. The “almost metric submersion” property implies that if $\delta$ is small enough then for all large $i$, we have

[TABLE]

for all $v\in H_{u_{i}}$. It follows that for $\eta\in\Lambda^{p}(T_{u}^{*}B(0,\delta r_{\delta}))$, we have

[TABLE]

Lemma 4.25**.**

There is some $\mathaccent 866{C}<\infty$ so that for all sufficiently large $i$ and all $u,u^{\prime}\in B(0,\delta r_{\delta})$, we have

[TABLE]

Proof.

The proof uses gradient flow. We sketch the argument, which is similar to [18, Pf. of Lemma 6.15]. Write $\tau_{i}=(\xi_{i,1},\ldots,\xi_{i,n_{\infty}})$ and $u=(a_{1},\ldots,a_{n_{\infty}})$. From the construction of $\tau_{i}$ using distance functions from strainer points, the functions $\xi_{i,\cdot}$ are quantitatively semiconcave, independent of $i$. Put $H_{-}(u)=\{\mathaccent 866{u}=(b_{1},\ldots,b_{n_{\infty}})\in{\mathbb{R}}^{n_{\infty}}:b_{l}\leq a_{l}\mbox{ for }1\leq l\leq n_{\infty}\}$. Given $m_{i}\in\tau_{i}^{-1}(u^{\prime})$, we can perform a gradient flow starting from $m_{i}$ with respect to the gradient of the distance function from an appropriate point in $M_{i}$, for a controlled amount of time, to ensure that the result lies in $\tau_{i}^{-1}(H_{-}(u))$. Then we perform a gradient flow with respect to the gradient of $F=\min(0,\xi_{i,1}-a_{1},\ldots,\xi_{i,n_{\infty}}-a_{n_{\infty}})$. After a controlled amount of time, the result of the flow lies in $\tau_{i}^{-1}(u)$. This gives a map $L_{u^{\prime},u}:\tau_{i}^{-1}(u^{\prime})\rightarrow\tau_{i}^{-1}(u)$. Using the control on the semiconcavity of the distance functions and of $F$, along with the ensuing distortion bounds for gradient flow, we obtain a bound on the Lipschitz constant of $L_{u^{\prime},u}$ that is independent of $u$, $u^{\prime}$ and $i$. Replacing $u^{\prime}$ by a point $u^{\prime\prime}$ moving along a line from $u^{\prime}$ to $u$, and performing the same construction, shows that if the fibers are orientable then $L_{u^{\prime},u}$ has degree one; if the fibers are not orientable then we pass to orientable double covers of the fibers and apply the same arguments. In all, from the Lipschitz bound on $L_{u^{\prime},u}$, we obtain a bound $\frac{\operatorname{vol}(\tau_{i}^{-1}(u))}{\operatorname{vol}(\tau_{i}^{-1}(u^{\prime}))}\leq\mathaccent 866{C}$ with $\mathaccent 866{C}$ independent of $u$, $u^{\prime}$ and $i$, thereby giving the first inequality in (4.26). Reversing the roles of $u$ and $u^{\prime}$ gives the second inequality in (4.26). ∎

We continue with the proof of Proposition 4.19. Let $\mu_{\delta}:U_{\delta}\rightarrow(0,1)^{n_{\infty}}$ be linear coordinates for a neighborhood $U_{\delta}$ of [math] in $B(0,\delta r_{\delta})$. We redefine $U_{i}$ to be $\tau_{i}^{-1}(U_{\delta})$ and put $h_{i}=\mu_{\delta}\circ\tau_{i}:U_{i}\rightarrow(0,1)^{n_{\infty}}$.

Lemma 4.27**.**

There exist $\Delta<\infty$ and, for sufficiently large $i$, a closed element $\chi_{i}\in\Omega^{n-n_{\infty}}_{\operatorname{Lip}}(U_{i};{\mathcal{O}}_{U_{i}})$ so that for all $u\in(0,1)^{n_{\infty}}$, we have $\int_{h_{i}^{-1}(u)}\chi_{i}=1$, and $\int_{h_{i}^{-1}(u)}|\chi|^{2}\operatorname{dvol}_{h_{i}^{-1}(u)}\leq\Delta\left(\inf_{u\in(0,1)^{n_{\infty}}}\operatorname{vol}(h_{i}^{-1}(u))\right)^{-1}$.

Proof.

We can assume that $h_{i}$ is similarly defined on a slightly larger open set containing $U_{i}$, so that the fiber $h_{i}^{-1}(1,\ldots,1)$ is well defined. Write $h_{i}=(h_{i,1}.\ldots,h_{i,n_{\infty}})$. Similarly to the proof of Lemma 4.25, we perform a gradient flow on $\overline{U_{i}}$ with respect to the gradient of $F=\min(0,h_{i,1}-1,\ldots,h_{i,n_{\infty}}-1)$. After a controlled amount of time, the result of the flow lies in $h_{i}^{-1}(1,\ldots,1)$. This gives a deformation retraction $L:\overline{U_{i}}\rightarrow h_{i}^{-1}(1,\ldots,1)$. Using the control on the semiconcavity of $F$, along with the ensuing distortion bounds for gradient flow, we obtain a bound on the Lipschitz constant of $F$. Let $\chi_{i}$ be the pullback under $L$ of the normalized volume density

[TABLE]

Then $\chi_{i}\in\Omega^{n-n_{\infty}}_{\operatorname{Lip}}(U_{i};{\mathcal{O}}_{U_{i}})$. The bound on the Lipschitz constant of $F$, which is independent of $i$, gives a pointwise bound on $\chi_{i}$ of the form $|\chi_{i}|\leq\operatorname{const.}\left(\operatorname{vol}(h_{i}^{-1}(1,\ldots,1))\right)^{-1}$. As

[TABLE]

the lemma follows. ∎

Define $\psi,E_{\psi},\operatorname{Ker}(d_{(0,1)^{n_{\infty}}})$ and $N_{\psi}$ as in the proof of Proposition 4.7, with $n$ replaced by $n_{\infty}$.

Let $R_{i}:(0,2^{-\frac{n!}{n_{\infty}}c_{i}})^{n_{\infty}}\rightarrow(0,1)^{n_{\infty}}$ be multiplication by $2^{\frac{n!}{n_{\infty}}c_{i}}$. Under rescaling,

[TABLE]

and the square norm of $R_{i}^{*}\psi$ in $\Omega^{p}_{L^{2}}((0,2^{-\frac{n!}{n_{\infty}}c_{i}})^{n_{\infty}})/\operatorname{Ker}(d)$ is $2^{\frac{n!}{n_{\infty}}c_{i}(2p-n_{\infty})}N_{\psi}$.

There are $\overline{k}_{i}=2^{n!c_{i}}$ disjoint boxes $\{B_{j}\}_{j=1}^{\overline{k}_{i}}$ in $(0,1)^{n_{\infty}}$, each congruent to $(0,2^{-\frac{n!}{n_{\infty}}c_{i}})^{n_{\infty}}$. Let $\psi_{j}$ be the translate of $R_{i}^{*}\psi$ to $B_{j}$. Let $V^{\infty}_{i}$ be the span of $\{\psi_{j}\}_{j=1}^{\overline{k}_{i}}$ in $\Omega^{p}_{L^{2}}\left((0,1)^{n_{\infty}}\right)$. Let ${h_{i}^{*}V^{\infty}_{i}}$ denote the extension by zero from $\Omega^{p}_{L^{2}}(U_{i})$ to $\Omega^{p}_{L^{2}}(M_{i})$ of the pullback.

Lemma 4.30**.**

For large $i$, $h_{i}^{*}V^{\infty}_{i}$ is a subspace of $\Omega^{p}_{L^{2},d}(M_{i})$ that projects to a $\overline{k}_{i}$-dimensional subspace $\mathaccent 1373{h_{i}^{*}V^{\infty}_{i}}$ of $\Omega^{p}_{L^{2},d}(M_{i})/\operatorname{Ker}(d_{M_{i}})$.

Proof.

We claim that $h_{i}^{*}\psi_{j}$ is a well-defined element of $\Omega^{p}_{L^{2},d}(M_{i})$, with $dh_{i}^{*}\psi_{j}=h_{i}^{*}d\psi_{j}$. Let $\Omega^{*}_{\operatorname{Lip},c}\left((0,1)^{n_{\infty}}\right)$ denote the differential graded algebra constructed from compactly supported Lipschitz functions on $(0,1)^{n_{\infty}}$. There is a pullback $h_{i}^{*}:\Omega^{*}_{\operatorname{Lip},c}\left((0,1)^{n_{\infty}}\right)\rightarrow\Omega^{*}_{\operatorname{Lip}}(M_{i})$ of differential graded algebras. Since $\psi_{j}$ is a smooth compactly supported $p$-form on $(0,1)^{n_{\infty}}$, it follows as in the proof of Lemma 2.7 that there is a $\psi_{j}^{\prime}\in\Omega^{*}_{\operatorname{Lip},c}\left((0,1)^{n_{\infty}}\right)$ so that $\rho(\psi_{j}^{\prime})=\psi_{j}$. Then $h_{i}^{*}\psi_{j}=\rho(h_{i}^{*}\psi_{j}^{\prime})$ in $\Omega^{p}_{L^{2}}(M_{i})$. Using Lemma 2.2, one shows that $h_{i}^{*}\psi_{j}$ lies in $\Omega^{p}_{L^{2},d}(M_{i})$, with differential given by $dh_{i}^{*}\psi_{j}=\rho(dh_{i}^{*}\psi_{j}^{\prime})=\rho(h_{i}^{*}d\psi_{j}^{\prime})=h_{i}^{*}d\psi_{j}$.

The lemma now follows as in the proof of Proposition 4.7. ∎

For sufficiently small $\delta$ and all large $i$, the ratio $\frac{(h_{i})_{*}\operatorname{dvol}_{M_{i}}}{\operatorname{dvol}_{(0,1)^{n_{\infty}}}}$ is bounded above by twice the function on $(0,1)^{n_{\infty}}$ which, to a point $u\in(0,1)^{n_{\infty}}$, assigns the volume of the fiber $h_{i}^{-1}(u)$. If $\eta_{i}=\sum_{j=1}^{\overline{k}_{i}}\beta_{j}h_{i}^{*}\psi_{j}$ is a nonzero element of $h_{i}^{*}V^{\infty}_{i}$ then using (4.24), there is some $\Lambda<\infty$ such that for large $i$, we have

[TABLE]

Let $\mathaccent 869{\eta}_{i}$ be the class of $\eta_{i}$ in $\mathaccent 1373{h_{i}^{*}V^{\infty}_{i}}$. We claim that

[TABLE]

To see this, suppose that $\|\mathaccent 869{\eta}_{i}\|^{2}<\frac{1}{4}2^{\frac{n!}{n_{\infty}}c_{i}(2p-n_{\infty})}\Delta^{-1}N_{\psi}\left(\inf_{u\in(0,1)^{n_{\infty}}}\operatorname{vol}(h_{i}^{-1}(u))\right)\sum_{j=1}^{\overline{k}_{i}}|\beta_{j}|^{2}$. Then there is some $\sigma_{i}\in\operatorname{Ker}(d_{M_{i}})$ so that

[TABLE]

Let $\int_{Z_{i}}:\Omega^{*}_{L^{2}}\left(U_{i};{\mathcal{O}}_{i}\right)\rightarrow\Omega^{*-(n-n_{\infty})}_{L^{2}}\left((0,1)^{n_{\infty}}\right)$ denote fiberwise integration.

Lemma 4.34**.**

The form $\int_{Z_{i}}\chi_{i}\wedge\sigma_{i}\in\Omega^{p}_{L^{2}}\left((0,1)^{n_{\infty}}\right)$ lies in $\operatorname{Ker}(d_{(0,1)^{n_{\infty}}})$.

Proof.

Given a compactly supported $\omega^{\prime}\in\Omega^{n_{\infty}-p-1}_{\operatorname{Lip}}\left((0,1)^{n_{\infty}}\right)$, we have

[TABLE]

Smoothing $\sigma_{i}$ by applying the heat operator on $M_{i}$, and using Lemma 2.2, gives

[TABLE]

The lemma follows. ∎

Put $\nu_{i}=\int_{Z_{i}}\chi_{i}\wedge\sigma_{i}$. As

[TABLE]

it follows that

[TABLE]

Using Lemma 4.27 and (4.33),

[TABLE]

However,

[TABLE]

This is a contradiction.

To finish the proposition, combining Lemma 4.25, (4.31) and (4.32) gives

[TABLE]

For large $i$, this contradicts the fact that $\lambda_{\overline{k}_{i},p}(M_{i})\geq\frac{i}{4^{\frac{n!}{s}}}\overline{k}_{i}^{\frac{2}{s}}$. ∎

Theorem 1.6 follows from Proposition 4.19 and the Hodge decomposition.

Remark 4.42*.*

In addition to upper bounds on the eigenvalues, one could ask about lower bounds. Under the assumption of a lower bound on the curvature operator, there are lower bounds on $\lambda_{k,p}$ in terms of the diameter, coming from heat trace estimates, with the bound proportionate to $k^{\frac{2}{n}}$ as $k\rightarrow\infty$ [1]. (For some finite number of $k$’s, the lower bound may be zero.) If we only assume a lower bound on the sectional curvature then it is not so clear if there are eigenvalue bounds from below, in terms of the diameter. As a consistency check, we note that eigenvalue bounds from below, going to infinity as $k\rightarrow\infty$, imply upper bounds on Betti numbers. Such upper bounds on Betti numbers exist if we assume a lower bound on the curvature operator or, more generally a lower bound on sectional curvatures. However, the proofs are very different in the two cases. With a lower bound on the curvature operator, a Betti number bound (in terms of the diameter) follows easily from heat trace estimates. In comparison, if we assume a lower bound on sectional curvatures then there is again a Betti number bound, but the proof uses completely different methods [13].

One can also ask about spectral convergence. That is, suppose that $\{(M_{i},g_{i})\}_{i=1}^{\infty}$ is a sequence of Riemannian manifolds satisfying the assumptions of Theorem 1.3, with a Gromov-Hausdorff limit $X$. The question is whether $\lim_{i\rightarrow\infty}\lambda_{k,p}(M_{i})=\lambda_{k,p}(X)$. This is known for functions when the lower sectional curvature bound is replaced by a lower Ricci curvature bound [5], and for $1$-forms when the lower sectional curvature bound is replaced by a double sided Ricci bound [17]. It may be necessary to assume that the Riemannian manifolds $\{(M_{i},g_{i})\}_{i=1}^{\infty}$ have a uniform lower bound on the curvature operator.

5. Hodge theorem and compact resolvent

In this section we introduce the class ${\mathcal{C}}_{*}$ of Lipschitz multiconical spaces. If $X\in{\mathcal{C}}_{*}$ then in Subsection 5.1 we prove a Hodge theorem, in the sense that we identify $\operatorname{Ker}(\triangle_{*})$ with a certain intersection homology group. In Subsection 5.2 we show that $(I+\triangle_{*})^{-1}$ is compact.

To begin, if $Y$ is a metric space of diameter at most $\pi$, and $\epsilon>0$, then the truncated open metric cone $CY(\epsilon)$ over $Y$ is homeomorphic to the topological space $([0,\epsilon)\times Y)/\sim$, where $(0,y_{1})\sim(0,y_{2})$ for all $y_{1},y_{2}\in Y$. The vertex of the cone, i.e. the equivalence class $\{(0,y)\}_{y\in Y}$, is denoted by $\star$. The metric on $CY$ comes from

[TABLE]

We define a class ${\mathcal{C}}_{*}$ of compact metric spaces inductively. An element of ${\mathcal{C}}_{0}$ is a finite metric space. A compact metric space $X$ lies in ${\mathcal{C}}_{n}$ if every point $x\in X$ has a neighborhood $U$ so that there is a pointed biLipschitz homeomorphism $h:(U,x)\rightarrow(CY(\epsilon),\star)$ for some $\epsilon>0$ and some $Y\in{\mathcal{C}}_{n-1}$ with $\operatorname{diam}(Y)\leq\pi$.

Note that for any $\epsilon>0$, the cone $CY(\epsilon)$ is biLipschitz homeomorphic to $CY(1)$, so the parameter $\epsilon$ is not really needed in the definition.

Example 5.2*.*

If $X$ is an $n$-dimensional closed Riemannian Lipschitz manifold then $X\in{\mathcal{C}}_{n}$.

By induction, if $X\in{\mathcal{C}}_{n}$ then there is an open dense subset $X^{*}$ of full Hausdorff $n$-measure that has the structure of a Riemannian Lipschitz $n$-manifold. Namely if $n=0$ then $X^{*}=X$. If $n>0$, cover $X$ by a finite number of neighborhoods $\{U_{i}\}_{i=1}^{N}$ of points $\{x_{i}\}_{i=1}^{N}$ so that there are pointed biLipschitz homeomorphisms $\phi_{i}:(U_{i},x_{i})\rightarrow(CY_{i}(\epsilon_{i}),\star_{i})$, with $Y_{i}\in{\mathcal{C}}_{n-1}$. Then we can take $X^{*}=\bigcup_{i=1}^{N}\phi_{i}^{-1}(CY_{i}^{*}(\epsilon_{i})-\star_{i})$.

We can apply the setup of Section 2.

Example 5.3*.*

An element $X$ of ${\mathcal{C}}_{1}$ is a finite metric graph $G$. The subset $X^{*}$ is the union of the open edges. For convenience, suppose that $X^{*}$ is oriented. Given $f\in\Omega^{0}_{L^{2}}(X)$ and an edge $e$ of length $L_{e}$, write f\Big{|}_{e}=f_{e}, where $f_{e}\in L^{2}(0,L_{e})$. Then $f\in\operatorname{Dom}(d)$ if

1. (1)

Each $f_{e}\in H^{1}(0,L_{e})$, and 2. (2)

For each vertex $v$ of $G$, the sum of the limiting values of $f_{e}$ along edges $e$ incoming to $v$ equals the sum of the limiting values of $f_{e}$ along edges outgoing from $v$.

Note that $\Omega^{0}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}=\Omega^{0}_{L^{2}}(X)$. A $1$-form $\omega\in\Omega^{1}_{L^{2},d}(X)=\Omega^{1}_{L^{2}}(X)$ consists of a union of $L^{2}$-regular $1$-forms on the open edges. The $L^{2}$-cohomology of $X$ is given in degree zero by $\operatorname{Ker}(d:\Omega^{0}_{L^{2},d}(X)\rightarrow\Omega^{1}_{L^{2},d}(X))\cong{\mathbb{R}}^{b_{1}(G)}$, and in degree one by $\Omega^{1}_{L^{2},d}(X)/{\operatorname{Im}(d)}\cong{\mathbb{R}}^{b_{0}(G)}$.

Given $\omega\in\Omega^{1}_{L^{2}}(X)$ and an edge $e$ of length $L_{e}$, write \omega\Big{|}_{e}={\omega}_{e}ds, where ${\omega}_{e}\in L^{2}(0,L_{e})$ and $s$ is the oriented length parameter along $e$. Then $\omega\in\operatorname{Dom}(d^{*})$ if

1. (1)

Each $\omega_{e}$ lies in $H^{1}(0,L_{e})$, and 2. (2)

For each vertex $v\in G$, there is a number $F_{v}$ so that for each edge $e$ adjoining $v$, the limiting value of $\omega_{e}$ on $e$, toward $v$, is $\pm F_{v}$, depending on whether $e$ is incoming or outgoing.

If $\omega\in\operatorname{Dom}(d^{*})$ then the restriction of $d^{*}\omega\in\Omega^{0}_{L^{2}}(X)$ to an edge $e$ is $-\>\frac{d\omega_{e}}{ds}$.

Then $\operatorname{Dom}(\triangle_{0})=\{f\in\Omega^{0}_{L^{2}}(X)\>:\>df\in\operatorname{Dom}(d^{*})\}$. The restriction of $\triangle_{0}f$ to an edge $e$ is $-\>\frac{d^{2}f_{e}}{ds^{2}}$. The operator $\triangle_{1}$ on $\Omega^{1}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$ vanishes.

To see how the orientation of $G$ affects the calculations, suppose that $G_{e}^{\prime}$ is the oriented graph obtained by starting with $G$ and reversing the orientation of a particular edge $e$. Given $f$ in the domain of $d$ for $G$, define $f^{\prime}$ by

[TABLE]

Then $f^{\prime}$ is in the domain of $d$ for $G^{\prime}$, and has the same energy as $f$. Hence if $f$ is in the domain of $\triangle_{0}$ for $G$, then $f^{\prime}$ is in the domain of $\triangle_{0}$ for $G^{\prime}$. It follows that choosing different orientations of the graph gives unitarily equivalent representations of $\triangle_{0}$.

5.1. Hodge theorem

For background information on intersection homology, we refer to [20].

Proposition 5.5**.**

If $X\in{\mathcal{C}}_{n}$ then for all $p\in[0,n]$, the unreduced $L^{2}$-cohomology

[TABLE]

is isomorphic to $\operatorname{IH}^{GM}_{n-p}(X;{\mathcal{O}})$. Here $\operatorname{IH}^{GM}_{*}$ denotes the Goresky-MacPherson intersection homology with real coefficients, computed with the perversity $\overline{p}$ given by $\overline{p}(0)=0$ and $\overline{p}(j)=\left[\frac{j-1}{2}\right]$ for $j\geq 1$, and ${\mathcal{O}}$ is the orientation line bundle of the codimension-zero stratum of $X$.

Proof.

We first note that $\operatorname{IH}$ is a topological invariant of $X$ [19, Theorem 9] but can be computed using a topological stratification. In our case there is a natural stratification $X=X_{n}\supset X_{n-1}\supset\ldots\supset X_{0}\supset X_{-1}=\emptyset$ given by saying that $x\in X_{j}$ if and only if there is no neighborhood of $x$ that is biLipschitz equivalent to $B^{j+1}\times CY^{n-j-2}$ for any $Y^{n-j-2}\in{\mathcal{C}}_{n-j-2}$ of diameter at most $\pi$. Here $B^{j+1}$ is the open unit ball in ${\mathbb{R}}^{j+1}$. The associated codimension-$k$ stratum is $X_{n-k}-X_{n-k-1}$, a manifold of dimension $n-k$. A point in the codimension-$k$ stratum has a neighborhood that splits off a $B^{k}$-factor, but there is no neighborhood that splits off a $B^{k+1}$-factor.

If ${\mathcal{O}}$ is the orientation line bundle of the codimension-zero stratum then since $\overline{p}(1)=0$, it follows as in [12, Section 2.2] that perversity-$\overline{p}$ intersection homology with values in ${\mathcal{O}}$ is well-defined.

If $X$ is a pseudomanifold, i.e. if $X_{n-1}=X_{n-2}$, then the result of Proposition 5.5 is well known and goes back to work of Cheeger [4] and Cheeger-Goresky-MacPherson [6]. The proofs that are relevant for us are sheaf-theoretic in nature, and appear in [25] and [36].

There are two relevant sheaves of differential graded complexes on $X$. The first one is the sheaf $\Omega^{*}_{L^{2}_{loc},d}$ defined in Section 3. We will use the fact that it only depends on $X$ through the biLipschitz homeomorphism class of $X$. The second relevant sheaf, as pointed out to me by Greg Friedman, is the sheaf $\operatorname{IC}_{n-\star}$ coming from the presheaf whose sections, over an open set $U\subset X$, are the singular intersection chains $C_{n-\star}(X,X-\overline{U};{\mathcal{O}})$ relative to the perversity $\overline{p}$ [10, Section 3]. (If $X$ is a PL-stratified space then $IC_{n-\star}$ is derived isomorphic to the sheaf $IC^{\infty}_{n-\star}$ whose sections, over an open set $U$, are the locally finite ${\mathcal{O}}$-valued simplicial intersection chains relative to $\overline{p}$.) The hypercohomology groups of the two sheaves are isomorphic to ${\mathcal{H}}^{*}_{L^{2}}(X)$ and $\operatorname{IH}^{GM}_{n-*}(X;{\mathcal{O}})$, respectively. Hence it suffices to show that the two sheaves are isomorphic in the derived category of differential graded sheaves on $X$.

When $X$ is a pseudomanifold, the strategy of [25] was to use the unique extension result of [12]. On the codimension-zero stratum, each sheaf was quasi-isomorphic to the constant ${\mathbb{R}}$-sheaf in degree [math]. As each sheaf satisfied the axioms of [12, Section 3.4], the stratum-by-stratum argument of [12, Section 3.5] showed that the two extensions from the codimension-zero stratum to all of $X$ are isomorphic in the derived category.

When the codimension-one stratum is nonempty, this strategy has to be slightly modified. The relevant unique extension result for us is in [14, Section 4], with $c_{p}=c=2$. We have to know that the restrictions of the two sheaves, on the union of the codimension-zero and codimension-one strata, are equivalent coefficient systems in the sense of [14, Definition 5.1]. We also have to know that the conditions of [14, Proposition 5.2(1)] are satisfied. Then [14, Proposition 4.5] implies that the two sheaves are isomorphic in the derived category of differential graded sheaves on $X$.

As the steps are similar to those in the pseudomanifold case, we just give the main points. We let $\operatorname{H}^{*}_{L^{2},d}(\cdot)$ denote hypercohomology of the sheaf $\Omega^{*}_{L^{2}_{loc},d}$ and we let $\operatorname{H}^{*}_{L^{2},d,c}(\cdot)$ denote compactly-supported hypercohomology. Similarly, we let $\operatorname{IC}^{GM}_{n-\star}(\cdot)$ denote hypercohomology of the sheaf $\operatorname{IC}_{n-\star}$ and we let $\operatorname{IC}^{GM}_{n-\star,c}(\cdot)$ denote compactly supported hypercohomology, i.e the usual ${\mathcal{O}}$-twisted Goresky-MacPherson intersection homology in degree $n-\star$.

First, we give the relevant cohomology of the truncated open metric cone $CY=CY(1)$ over some $Y\in{\mathcal{C}}_{k-1}$. If $k=1$ then

[TABLE]

where $\mathaccent 869{\operatorname{H}}$ denotes reduced homology, and

[TABLE]

If $k>1$ then

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Equations (5.9) and (5.10) can be proved using separation of variables as in [36]. Equation (5.9) can be understood as saying that the $L^{2}$-cohomology of $CY$ comes from pulling back harmonic forms from $Y$ with respect to the projection map $(0,1)\times Y\rightarrow Y$, provided that the pullback is square integrable. Equation (5.10) can be understood as saying that the compactly supported $L^{2}$-cohomology of $CY$ is generated by forms of the type $\phi^{\prime}dr\wedge\omega$, where $\phi\in C^{\infty}(0,1)$ is a nonincreasing function that is identically one on $(0,1/3)$ and identically zero on $(2/3,1)$, and $\omega$ is a harmonic $(i-1)$-form on $Y$. Then when $i\geq\frac{k}{2}+1$, the putative primitive $\phi\omega$ fails to be square integrable on $CY$. Equations (5.11) and (5.12) follow from [10, Proposition 2.18].

Next, for both sheaf cohomology theories there are Künneth formulas :

[TABLE]

The third isomorphism comes from [10, Proposition 2.20].

To check the conditions of [14], let $A$ denote one of the two above differential graded sheaves. They are both cohomologically constructible with respect to the stratification. As in [12, Section 1.12], if $v$ is the vertex of a cone $CY$, and $f:(0,v)\rightarrow B^{n-k}\times CY$ is inclusion, then $\operatorname{H}^{i}(f^{*}A)\cong\operatorname{H}^{i}(B^{n-k}\times CY;A)$ and $\operatorname{H}^{i}(f^{!}A)\cong\operatorname{H}_{c}^{i}(B^{n-k}\times CY;A)$. We will use the fact that the cohomology of $\Omega^{*}_{L^{2}_{loc},d}$ is biLipschitz invariant, in order to compute the $L^{2}$-cohomology of a neighborhood of a point $x\in X$ that is biLipschitz to $B^{n-k}\times CY$, using the conical metric on $CY$ and the product metric on $B^{n-k}\times CY$.

The restrictions of $\Omega^{*}_{L^{2}_{loc},d}$ and $\operatorname{IC}_{n-\star}$ to $X_{n}-X_{n-2}$ are quasi-isomorphic; c.f. (5.7) and (5.8). Let ${\mathcal{E}}$ denote their common class in the derived category of differential graded sheaves on $X_{n}-X_{n-2}$. From (5.7), (5.8) and (5.13), if $x\in X_{n}-X_{n-2}$ then $\operatorname{H}^{i}({\mathcal{E}}_{x})=0$ for $i>0$ and $\operatorname{H}^{i}(f_{x}^{!}{\mathcal{E}})=0$ for $i<n$. It follows from [14, Proposition 5.2] that the coefficient system satisfies [14, Definition 5.1].

Finally, from (5.9), (5.11) and (5.13), we have

[TABLE]

if $Y\in{\mathcal{C}}_{k-1}$ and $i>\overline{p}(k)$. From (5.10), (5.12) and (5.13), we have

[TABLE]

if $Y\in{\mathcal{C}}_{k-1}$ and $i<n-\max(n-2-\overline{p}(k),0)$. Using [14, Lemma 4.6], the sheaves $\Omega^{*}_{L^{2}_{loc},d}$ and $\operatorname{IC}_{n-\star}$ satisfy the axioms $A_{\overline{p}}$ in the sense of [14, Definition 4.3]. From [14, Remark 4.2] and [14, Proposition 4.5], it follows that the two sheaves are derived isomorphic on $X$. This proves the proposition. ∎

Corollary 5.16**.**

If $X\in{\mathcal{C}}_{n}$ then for all $p\in[0,n]$, we have $\dim(\operatorname{Ker}(\triangle_{p}))<\infty$.

Corollary 5.17**.**

If $X\in{\mathcal{C}}_{n}$ then for all $p\in[0,n]$, we have $\overline{\operatorname{Im}(d)}=\operatorname{Im}(d)\subset\Omega^{p}_{L^{2}}(X)$.

In particular, $\operatorname{Ker}(\triangle_{*})\cong{\mathcal{H}}^{*}_{L^{2}}(X)$.

5.2. Compactness of the resolvent

Proposition 5.18**.**

For any $X\in{\mathcal{C}}_{n}$, and for any $p\in[0,n]$, the operator $(I+\triangle_{p})^{-1}$ is a compact operator on $\Omega^{p}_{L^{2}}(X)/\overline{\operatorname{Im}(d)}$.

Proof.

We will prove the proposition by induction on $n$. It is true if $n=0$.

Using Corollary 5.17, we have

[TABLE]

The restriction of $I+\triangle_{p}$ to $\Omega^{p}_{L^{2},d}(X)/\operatorname{Ker}(d)$ equals $I+d^{*}d$. Let $\Omega^{p}_{H^{1},d}(X)/\operatorname{Ker}(d)\subset\Omega^{p}_{L^{2}}(X)/\operatorname{Ker}(d)$ denote $\Omega^{p}_{L^{2},d}(X)/\operatorname{Ker}(d)$ with the Hilbert space norm $\|\omega\|_{H^{1}}^{2}=\|d\omega\|_{L^{2}}^{2}+\|\omega\|_{L^{2}}^{2}$. The map $d:\Omega^{p}_{H^{1},d}(X)/\operatorname{Ker}(d)\rightarrow\operatorname{Im}(d)$ is invertible. Since $\operatorname{Im}(d)$ is closed in $\Omega^{p+1}_{L^{2}}(X)$, the open mapping theorem implies that the inverse $d^{-1}:\operatorname{Im}(d)\rightarrow\Omega^{p}_{H^{1},d}(X)/\operatorname{Ker}(d)\subset\Omega^{p}_{L^{2}}(X)/\operatorname{Ker}(d)$ is $L^{2}$-bounded. As $(d^{*}d)^{-1}=d^{-1}\left(d^{-1}\right)^{*}$, it suffices to show that $d^{-1}$ is compact.

We will construct an approximate inverse of $d$. Given $x\in X$, let $V_{x}$ be a neighborhood of $x$ for which there is a biLipschitz homeomorphism $\phi_{x}:(V_{x},x)\rightarrow(CY_{x}(2),\star_{x})$ with $Y_{x}\in{\mathcal{C}}_{n-1}$ of diameter at most $\pi$. (Note that the parameter $2$ in $CY_{x}(2)$ can always be chosen.) Put $U_{x}=\phi_{x}^{-1}(CY_{x}(1))$. As $\{U_{x}\}_{x\in X}$ is an open covering of $X$, we can choose a finite subcovering $\{U_{i}\}_{i=1}^{N}$, where we write $U_{i}=U_{x_{i}}$. We write the corresponding biLipschitz homeomorphisms as $\phi_{i}\>:\>(V_{i},x_{i})\rightarrow(CY_{i}(2),\star_{i})$.

Let $\Omega^{*}_{L^{2}}(V_{i})$ be the square-integrable forms of $V_{i}$. Let $\Omega^{*}_{L^{2},d,abs}(V_{i})$ be the square-integrable elements of $\Omega^{*}_{L^{2}_{loc},d}(V_{i})$, the latter being defined in Section 3. In terms of the conical coordinate $r$ on $V_{i}$, an element of $\Omega^{*}_{L^{2}}(V_{i})$ can be written as $\omega=\omega_{0}(r)+dr\wedge\omega_{1}(r)$, with $\omega_{0}(r)\in\Omega^{*}_{L^{2}}(Y_{i})$ and $\omega_{1}(r)\in\Omega^{*-1}_{L^{2}}(Y_{i})$. If $\omega\in\Omega^{*}_{L^{2},d,abs}(V_{i})$ then $\omega_{0}(r)\in\Omega^{*}_{L^{2},d}(Y_{i})$, $\omega_{1}(r)\in\Omega^{*-1}_{L^{2},d}(Y_{i})$, and $d_{Y_{i}}\omega_{0}$ and $\partial_{r}\omega_{0}-d_{Y_{i}}\omega_{1}$ are square-integrable on $V_{i}$. Let $d_{i}:\Omega^{*}_{L^{2},d,abs}(V_{i})\rightarrow\Omega^{*+1}_{L^{2}}(V_{i})$ be the differential on $V_{i}$. From (5.9), the unreduced $L^{2}$-cohomology of $V_{i}$ is finite dimensional and so $\operatorname{Im}(d_{i})$ is closed in $\Omega^{p+1}_{L^{2}}(V_{i})$. As before, the map $d_{i}:\Omega^{p}_{L^{2},d,abs}(V_{i})/\operatorname{Ker}(d_{i})\rightarrow\operatorname{Im}(d_{i})$ has an inverse that extends to a bounded map $d_{i}^{-1}:\operatorname{Im}(d_{i})\rightarrow\Omega^{p}_{L^{2}}(V_{i})/\operatorname{Ker}(d_{i})$.

Lemma 5.20**.**

For each $i$, the operator $d_{i}^{-1}$ is compact.

Proof.

Under a biLipschitz change of metric on $V_{i}$, one obtains equivalent norms on $\operatorname{Im}(d_{i})$ and $\Omega^{p}_{L^{2}}(V_{i})/\operatorname{Ker}(d_{i})$. Hence to prove the lemma, we can replace $V_{i}$ by $CY_{i}(2)$ with its conical metric.

The quadratic form $Q_{i}$ on $\Omega^{p}_{L^{2},d_{i},abs}(CY_{i}(2))/\operatorname{Ker}(d_{i})$ is defined as in (2.19). The corresponding operator $d_{i}^{*}d_{i}$, densely defined on $\Omega^{p}_{L^{2}}(CY_{i}(2))/\operatorname{Ker}(d_{i})$, has a domain whose elements satisfy absolute (Neumann) boundary conditions on $Y_{i}=\partial\overline{CY_{i}(2)}$. The spectrum of $d_{i}^{*}d_{i}$ can be explicitly computed using separation of variables. If $Y_{i}$ is a smooth manifold then this was done in [15, 34]. (Since we are not interested in enforcing Hodge duality, subtleties about ideal boundary conditions do not arise.) In our case, by the induction assumption, the differential form Laplacian on $Y_{i}$ has a discrete spectrum with finite multiplicities. Then by the same separation of variable argument, this is also true for $d_{i}^{*}d_{i}$ on $CY_{i}(2)$. From the explicit spectral decomposition, one sees that $d_{i}^{-1}=(d_{i}^{*}d_{i})^{-1}d_{i}^{*}$ is a compact operator on $\operatorname{Im}(d_{i})$. ∎

There is a restriction map from $\operatorname{Im}(d)\subset\Omega^{*}_{L^{2}}(X)$ to $\operatorname{Im}(d_{i})\subset\Omega^{*}_{L^{2}}(V_{i})$. Let $L_{i}:\Omega^{*}_{L^{2}}(V_{i})/\operatorname{Ker}(d_{i})\rightarrow\operatorname{Ker}(d_{i})^{\perp}$ be the lifting isomorphism. Let $q:\Omega^{*}_{L^{2}}(X)\rightarrow\Omega^{*}_{L^{2}}(X)/\operatorname{Ker}(d)$ be the quotient map. Let $\{\sigma_{i}\}_{i=1}^{N}$ be a Lipschitz partition of unity subordinate to $\{U_{i}\}_{i=1}^{N}$. Define a compact operator $A:\operatorname{Im}(d)\rightarrow\Omega^{p}_{L^{2}}(X)/\operatorname{Ker}(d)$ by

[TABLE]

Then

[TABLE]

The operator $K$ given by

[TABLE]

is compact. As $dA=I+K$ and $d^{-1}$ is bounded, we see that $d^{-1}=A-d^{-1}K$ is compact. ∎

This proves Theorem 1.16.