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The moduli space of Riemann surfaces is Kahler hyperbolic

Curtis T. McMullen
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Abstract

Let $\cM_{g,n}$ be the moduli space of Riemann surfaces of genus $g$ with $n$ punctures. From a complex perspective, moduli space is hyperbolic. For example, $\cM_{g,n}$ is abundantly populated by immersed holomorphic disks of constant curvature -1 in the Teichmüller (=Kobayashi) metric. When $r=\dim_{\cx} \cM_{g,n}$ is greater than one, however, $\cM_{g,n}$ carries no complete metric of bounded negative curvature. Instead, Dehn twists give chains of subgroups $\zed^r \subset π_1(\cM_{g,n})$ reminiscent of flats in symmetric spaces of rank $r>1$. In this paper we introduce a new Kähler metric on moduli space that exhibits its hyperbolic tendencies in a form compatible with higher rank.

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Curtis T. McMullen. 2000-10-02. The moduli space of Riemann surfaces is Kahler hyperbolic. https://arxiv.org/abs/math/0010022

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