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arXiv · 1206.4010

Spectral theory for the Weil-Petersson Laplacian on the Riemann moduli space

Lizhen Ji·Rafe Mazzeo·Werner Müller·Andras Vasy
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Abstract

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar Laplacian is essentially self-adjoint, which then implies that its spectrum is discrete. The second theorem is a Weyl asymptotic formula for the counting function for this spectrum.

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Lizhen Ji, Rafe Mazzeo, Werner Müller, Andras Vasy. 2012-06-18. Spectral theory for the Weil-Petersson Laplacian on the Riemann moduli space. https://arxiv.org/abs/1206.4010

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