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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
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 \(\mathcal{H}_{\mathrm{ice}}(M)\simeq IH_{\overline{\mathrm{m}}}(M_{\mathrm{stra}})\).
These results are proved under the condition \(k\ge 3\) and a Witt type condition (\(Z\) is odd dimensional or with null cohomology in middle degree. Witt spaces have been introduced by P. H. Siegel [Am. J. Math. 105, 1067–1105 (1983; Zbl 0547.57019)]).
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.

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
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References:

[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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