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

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.