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

TL;DR

This paper investigates the spectral properties of the Weil-Petersson Laplacian on the moduli space of Riemann surfaces, proving essential self-adjointness and deriving a Weyl asymptotic formula for its spectrum.

Contribution

It establishes the essential self-adjointness of the Weil-Petersson Laplacian and derives a Weyl law for its spectral counting function on the moduli space.

Findings

Laplacian is essentially self-adjoint

Spectrum of the Laplacian is discrete

Weyl asymptotic formula for the spectrum

Abstract

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_\gamma$, the Riemann moduli space of surfaces of genus $\gamma > 1$. 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.