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

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Abstract

Let Mg,n{\mathcal M}_{g,n} be the moduli space of algebraic curves of genus gg with nn punctures, which is a noncompact orbifold. Let Mg,nDM\overline{\mathcal M}^{DM}_{g,n} denote its Deligne–Mumford compactification. Then Mg,n{\mathcal M}_{g,n} admits a class of canonical Riemannian and Finsler metrics. We probe the analogy between Mg,n{\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<1 < p < \infty, the LpL^p-cohomology of Mg,n{\mathcal M}_{g,n} with respect to these Riemannian metrics that are complete can be identified with the (ordinary) cohomology of Mg,nDM\overline{\mathcal M}^{DM}_{g,n}, and hence the LpL^p-cohomology is the same for different values of pp. This suggests a “rank-one nature” of the moduli space Mg,n{\mathcal M}_{g,n} from the point of view of LpL^p-cohomology. On the other hand, the LpL^p-cohomology of Mg,n{\mathcal M}_{g,n} with respect to the incomplete Weil Petersson metric is either the cohomology of Mg,nDM\overline{\mathcal M}^{DM}_{g,n} or that of Mg,n{\mathcal M}_{g,n} itself, depending on whether p43p\leq \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.