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

Lizhen Ji University of Michigan Rafe Mazzeo Stanford University Werner Müller Universität Bonn Andras Vasy Stanford University

Analysis of PDEs Differential Geometry mathscidoc:1705.03002

Distinguished Paper Award in 2017

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.
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@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.
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