# MathSciDoc: An Archive for Mathematician ∫

#### Analysis of PDEsDifferential Geometrymathscidoc:1705.03002

Comment. Math. Helv, 89, 867–894, 2014
We study the spectral geometric properties of the scalar Laplace–Beltrami operator associated to the Weil–Petersson metric $g_{WP}$ on $\mathcal{M}_{\gamma}$ , the Riemann moduli space of surfaces of genus  $\gamma>1$. This space has a singular compactification with respect to $g_{WP}$, 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.
Spectral theory,Weil–Petersson metric, moduli space, Riemann surface,Weyl law.
@inproceedings{lizhen2014spectral,
title={Spectral theory for the Weil–Petersson Laplacian on the Riemann moduli space},
author={Lizhen Ji, Rafe Mazzeo, Werner Müller, and Andras Vasy},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530155907957003770},
booktitle={Comment. Math. Helv},
volume={89},
pages={867–894},
year={2014},
}

Lizhen Ji, Rafe Mazzeo, Werner Müller, and Andras Vasy. Spectral theory for the Weil–Petersson Laplacian on the Riemann moduli space. 2014. Vol. 89. In Comment. Math. Helv. pp.867–894. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170530155907957003770.