[Paper Review] Harmonic forms on manifolds with edges

Eugénie Hunsicker, Rafe Mazzeo|ArXiv.org|Mar 16, 2005

Geometric Analysis and Curvature Flows11 references4 citations

TL;DR

This paper computes the space of $L^2$ harmonic forms on compact Riemannian stratified spaces with incomplete edge singularities, using weighted de Rham cohomology and edge pseudodifferential calculus. It establishes that the $L^2$ harmonic forms correspond to intersection cohomology groups when the fiber cohomology vanishes in critical degrees, resolving a key case in Hodge theory on singular spaces with incomplete metrics.

ABSTRACT

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$.

Motivation & Objective

• To determine the topological meaning of $L^2$ harmonic forms on compact stratified spaces with incomplete edge singularities.
• To extend Hodge theory to incomplete edge metrics by analyzing weighted $L^2$ cohomology and harmonic forms.
• To clarify the relationship between $L^2$ harmonic forms and intersection cohomology in the presence of singular strata.
• To resolve the dependence on boundary conditions at the singular locus for incomplete metrics by distinguishing minimal and maximal de Rham complexes.
• To characterize when the minimal and maximal $L^2$ de Rham complexes coincide, enabling identification of harmonic forms with intersection cohomology.

Proposed method

• Uses polynomially weighted $L^2$ cohomology to analyze differential forms on the regular part of the stratified space.
• Applies edge pseudodifferential calculus to construct parametrices for the Hodge Laplacian on incomplete edge metrics.
• Relies on the fibred cusp and fibred boundary calculus analogues, adapted to incomplete metrics via edge calculus.
• Analyzes the normal operator of the Laplacian to study spectral gaps and essential spectrum, particularly near zero.
• Employs sheaf-theoretic techniques to compute intersection cohomology, especially for cone bundles over the singular stratum.
• Distinguishes minimal and maximal extensions of the de Rham complex to handle the incomplete metric and singular boundary conditions.

Experimental results

Research questions

• RQ1 How do weighted $L^2$ harmonic forms on incomplete edge metrics relate to topological invariants of the underlying stratified space?
• RQ2 Under what conditions do the minimal and maximal de Rham complexes on the regular part of an incomplete edge space coincide?
• RQ3 When does the space of $L^2$ harmonic forms on an incomplete edge metric correspond to intersection cohomology?
• RQ4 What role does the cohomology of the fiber $F$ play in determining the dimensionality of $L^2$ harmonic forms when the base dimension $b$ is odd?
• RQ5 How does the essential spectrum of the Hodge Laplacian influence the existence of infinite-dimensional $L^2$ harmonic forms?

Key findings

• For incomplete edge metrics, $L^2$ harmonic $k$-forms decompose as $L^2 ilde{ ho}^k(M,g) imes L^2 ilde{ ho}^{k-r}(M,g)$ when the fiber cohomology $H^ ho(F)$ vanishes.
• When $b$ is even or $H^ ho(F) = 0$ for $ ho = (2k - b - 1)/2$, the minimal and maximal de Rham complexes coincide, ensuring $L^2$ harmonic forms are well-defined.
• The space of $L^2$ harmonic $k$-forms is isomorphic to the intersection cohomology ${I ilde{H}}^k_{rak{m}}(X)$ under the same cohomological vanishing condition.
• When $b$ is odd and $H^ ho(F) eq 0$, the space $L^2 ilde{ ho}^k(M,g)$ is infinite-dimensional due to the essential spectrum of $ riangle_k$ accumulating at zero.
• The spectral gap at zero for $ riangle_k$ is established via a parametrix construction in the edge calculus, ensuring the existence of a compact remainder for $ riangle_k - u$ when $ u$ is small and nonzero.
• The result generalizes Cheeger and Dai’s work by extending the correspondence to even-dimensional fibers and a full scale of polynomial weights.