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

### Lizhen Ji

University of Michigan, Ann Arbor, USA### Rafe Mazzeo

Stanford University, USA### Werner Müller

Universität Bonn, Germany### András Vasy

Stanford University, USA

## Abstract

We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $M_{γ}$, the Riemann moduli space of surfaces of genus $γ>1$. This space has a singular compactification with respect to $g_{WP}$, 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.

## Cite this article

Lizhen Ji, Rafe Mazzeo, Werner Müller, András Vasy, Spectral theory for the Weil–Petersson Laplacian on the Riemann moduli space. Comment. Math. Helv. 89 (2014), no. 4, pp. 867–894

DOI 10.4171/CMH/336