On the $L^p$-cohomology and the geometry of metrics on moduli spaces of curves

Abstract

Let ${\mathcal{M}}_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $n$ punctures, which is a noncompact orbifold. Let ${\overline{\mathcal{M}}}_{g,n}^{DM}$ denote its Deligne–Mumford compactification. Then ${\mathcal{M}}_{g,n}$ admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between ${\mathcal{M}}_{g,n}$ (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 $1<p<\infty$, the $L^p$-cohomology of ${\mathcal{M}}_{g,n}$ with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of ${\overline{\mathcal{M}}}_{g,n}^{DM}$, and hence the $L^p$-cohomology is the same for different values of $p$. This suggests a “rank-one nature” of the moduli space ${\mathcal{M}}_{g,n}$ from the point of view of $L^p$-cohomology. On the other hand, the $L^p$-cohomology of ${\mathcal{M}}_{g,n}$ with respect to the incomplete Weil Petersson metric is either the cohomology of ${\overline{\mathcal{M}}}_{g,n}^{DM}$ or that of ${\mathcal{M}}_{g,n}$ itself, depending on whether $p\le \frac{4}{3}$ or not. At the end of the chapter, we pose several natural problems on the geometry and analysis of these complete Riemannian metrics.