The moduli space of Riemann surfaces is Kahler hyperbolic

2000/10/02 by Curtis T. McMullen · 3 citations
Mathematics · #math.CV #msc:32Gxx

paper · pdf

published as Ann. of Math. (2) 151 (2000), no. 1, 327--357 · 31 pages

arxiv created 2000/10/02 · arxiv updated 2009/11/30

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 ⊂ π₁(\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.

Cited by

• Volume functions and boundary data of 3-dimensional hyperbolic manifolds
• Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves
• Regularity of harmonic maps into Teichmüller space

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