# On the $L_{p}$-cohomology and the geometry of metrics on moduli spaces of curves

### Lizhen Ji

University of Michigan, Ann Arbor, USA### Steven Zucker

The Johns Hopkins University, Baltimore, USA

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## Abstract

Let $M_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $n$ punctures, which is a noncompact orbifold. Let $M_{g,n}$ denote its Deligne–Mumford compactification. Then $M_{g,n}$ admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between $M_{g,n}$ (resp. Teichmüller spaces) with these metrics and certain noncompact locally symmetric spaces (resp. symmetric spaces of noncompact type) with their natural metrics. In this chapter, we observe that for all $1<p<∞$, the $L_{p}$-cohomology of $M_{g,n}$ with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of $M_{g,n}$, and hence the $L_{p}$-cohomology is the same for different values of $p$. This suggests a “rank-one nature” of the moduli space $M_{g,n}$ from the point of view of $L_{p}$-cohomology. On the other hand, the $L_{p}$-cohomology of $M_{g,n}$ with respect to the incomplete Weil Petersson metric is either the cohomology of $M_{g,n}$ or that of $M_{g,n}$ itself, depending on whether $p≤34 $ or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics.