Spectral and Hodge theory of “Witt” incomplete cusp edge spaces. (English) Zbl 1440.58013
Let \(M\) be a compact \(n\)-dimensional manifold, whose boundary \(\partial M\) is the total space of a fibration \(Z\to\partial M\xrightarrow{\pi} Y\), let \(x:M\to \mathbb{R}\) be a defining function of its boundary.
An incomplete cusp edge metric \(g_{\mathrm{ice}}\) on \(M\) has near the boundary \(\partial M=\{x=0\}\) the local form
\[
g_{\mathrm{ice}}=dx^2+x^{2k}g_Z+\pi^*g_Y+\widetilde g
\]
near the boundary, where \(k>1\), \(g_Y\) is a metric on \(Y\) and \(g_Z\) is positive definite when restricted to the fibers.
The Weil-Petersson (WP) metric on the compactified Riemann moduli space \(\mathcal{M}_{\gamma,\ell}\) (Riemann surfaces of genus \(\gamma\) and with \(\ell\) marked points) is an incomplete cusp edge metric (with \(k=3\)), see R. Mazzeo and J. Swoboda [Int. Math. Res. Not. 2017, No. 6, 1749–1786 (2017; Zbl 1405.32014)]; for the analysis on the Riemann moduli space, see the recent paper by K. Liu et al. [Pure Appl. Math. Q. 10, No. 2, 223–243 (2014; Zbl 1307.32012)]. Hodge theory results established in this paper for \(g_{\mathrm{ice}}\) metrics are hence valid for any WP metric. These results complete the spectral properties of the Laplacian operator on functions established by Ji et al. [Comment. Math. Helv. 89, No. 4, 867–894 (2014; Zbl 1323.35119)].
Let \(\Delta_{\mathrm{ice}}=d\delta+\delta g\) be the Hodge operator on smooth functions on the interior of \(M\) (in the case of the Riemann moduli space, the domain must be adapted to some orbifold singularities), let \(\mathcal{H}_{\mathrm{ice}}(M)= \{\alpha\in L^2(\Omega^*_{\mathrm{ice}}(M))|d\alpha=0, \delta\alpha=0\}\) be the harmonic \(L^2\) forms space and let \(IH_{\overline{\mathrm{m}}}(M_{\mathrm{stra}})\) be the middle perversity intersection cohomology of the stratified space \(M_{\mathrm{stra}}\) obtained by collapsing the fibration \(\partial M\) over \(Y\): \(M_{\mathrm{stra}}=M/\{p\sim q|p,q\in\partial M, \pi(p)=\pi(q)\}\).
The main results of the paper are
The main tool to achieve these results is the construction of the fundamental solution to the heat operator \(\mathrm{e}^{-t\Delta_{\mathrm{isc}}}\), whose kernel is defined on an appropriate blowup \(M^2_{\mathrm{heat}}\to M\times M\times \mathbb{R}\). The crucial, and very accurate, analysis of the heat kernel restriction to the boundary hypersurfaces is similar to the construction used by E. A. Mooers [J. Anal. Math. 78, 1–36 (1999; Zbl 0981.58022)]; it has been achieved in the geometric microlocal analysis framework set up by R. B. Melrose [Int. Math. Res. Not. 1992, No. 3, 51–61 (1992; Zbl 0754.58035)]. Bounds on the growth of \(L^2\) harmonic forms at the singular set are established before proving the Hodge theorems above.
The Weil-Petersson (WP) metric on the compactified Riemann moduli space \(\mathcal{M}_{\gamma,\ell}\) (Riemann surfaces of genus \(\gamma\) and with \(\ell\) marked points) is an incomplete cusp edge metric (with \(k=3\)), see R. Mazzeo and J. Swoboda [Int. Math. Res. Not. 2017, No. 6, 1749–1786 (2017; Zbl 1405.32014)]; for the analysis on the Riemann moduli space, see the recent paper by K. Liu et al. [Pure Appl. Math. Q. 10, No. 2, 223–243 (2014; Zbl 1307.32012)]. Hodge theory results established in this paper for \(g_{\mathrm{ice}}\) metrics are hence valid for any WP metric. These results complete the spectral properties of the Laplacian operator on functions established by Ji et al. [Comment. Math. Helv. 89, No. 4, 867–894 (2014; Zbl 1323.35119)].
Let \(\Delta_{\mathrm{ice}}=d\delta+\delta g\) be the Hodge operator on smooth functions on the interior of \(M\) (in the case of the Riemann moduli space, the domain must be adapted to some orbifold singularities), let \(\mathcal{H}_{\mathrm{ice}}(M)= \{\alpha\in L^2(\Omega^*_{\mathrm{ice}}(M))|d\alpha=0, \delta\alpha=0\}\) be the harmonic \(L^2\) forms space and let \(IH_{\overline{\mathrm{m}}}(M_{\mathrm{stra}})\) be the middle perversity intersection cohomology of the stratified space \(M_{\mathrm{stra}}\) obtained by collapsing the fibration \(\partial M\) over \(Y\): \(M_{\mathrm{stra}}=M/\{p\sim q|p,q\in\partial M, \pi(p)=\pi(q)\}\).
The main results of the paper are
- ●
- For each integer \(d=0,\dots,n\), the Hodge Laplacian \(\Delta^d_{\mathrm{ice}}\) on \(d\)-forms is essentially self-adjoint and has discrete spectrum \((\lambda_{j,d}),j\in\mathbb{N}\), with Weyl asymptotics \(\#\{j|\lambda_{j,d}<\lambda\}=_{\lambda\to\infty}(2\pi)^{-n}v_n\mathrm{vol}(M_{\mathrm{ice}})\lambda^n(1+o(1))\) where \(v_n\) is the volume of the \(n\) dimensional unit euclidean ball.
- ●
- The \(L^2\) harmonic forms obey \(\mathcal{H}_{\mathrm{ice}}(M)= \{\alpha\in L^2(\Omega^*_{\mathrm{ice}}(M))|\Delta_{\mathrm{ice}}\alpha=0\}\).
- ●
- There is a natural isomorphism \(\mathcal{H}_{\mathrm{ice}}(M)\simeq IH_{\overline{\mathrm{m}}}(M_{\mathrm{stra}})\).
The main tool to achieve these results is the construction of the fundamental solution to the heat operator \(\mathrm{e}^{-t\Delta_{\mathrm{isc}}}\), whose kernel is defined on an appropriate blowup \(M^2_{\mathrm{heat}}\to M\times M\times \mathbb{R}\). The crucial, and very accurate, analysis of the heat kernel restriction to the boundary hypersurfaces is similar to the construction used by E. A. Mooers [J. Anal. Math. 78, 1–36 (1999; Zbl 0981.58022)]; it has been achieved in the geometric microlocal analysis framework set up by R. B. Melrose [Int. Math. Res. Not. 1992, No. 3, 51–61 (1992; Zbl 0754.58035)]. Bounds on the growth of \(L^2\) harmonic forms at the singular set are established before proving the Hodge theorems above.
Reviewer: Laurent Guillopé (Nantes)
MSC:
58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |
58J20 | Index theory and related fixed-point theorems on manifolds |
58A35 | Stratified sets |
55N33 | Intersection homology and cohomology in algebraic topology |
32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
35K08 | Heat kernel |
35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
35R01 | PDEs on manifolds |
47A10 | Spectrum, resolvent |
58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
58J37 | Perturbations of PDEs on manifolds; asymptotics |
Keywords:
Hodge theory; singular spaces; Hodge-Laplace operator; heat equation; \(L^2\)-cohomology; intersection cohomology; Riemann moduli space; stratified space; spectral theory; Weil-Petersson metric; Weyl asymptoticsCitations:
Zbl 1405.32014; Zbl 1307.32012; Zbl 1323.35119; Zbl 0547.57019; Zbl 0981.58022; Zbl 0754.58035References:
[1] | 762 702J. Gell-Redman and J. SwobodaCMH 1. Introduction On a compact manifoldMwith boundary@Mwhich is the total space of a fiber bundle Z ,!@M!Y;(1.1) |
[2] | @M in order to obtain a self-adjoint operator. Theorem 1.1.Let.M; gice/be an incomplete cusp edge manifold that is “Witt,” meaning that eitherdimZDfis odd or Hf =2.Z/D f0g:(1.4) |
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