The moduli space of Riemann surfaces is Kähler hyperbolic

By Curtis T. McMullen. 2000

158 citations recorded by the source corpus.

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 Teichm\"uller (=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 $\zed^r \subset \pi_1(\cM_{g,n})$ reminiscent of flats in symmetric spaces of rank $r>1$. In this paper we introduce a new K\"ahler metric on moduli space that exhibits its hyperbolic tendencies in a form compatible with higher rank.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2178c44e812907d7d7dea59e984ecc8e112d8931

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Geometry of the Weil-Petersson completion of Teichm\ (2005, background): “As applications McMullen found: a positive lower bound for the Teichmüller Rayleigh-Ritz quotient, a complex submanifold isoperimetric inequality, and the alternating sign of the orbifold Euler characteristic for the moduli spaces [32].”
Geometry of domains with the uniform squeezing property (2009, methodology): “It is also proved in [ Mc ] that any order of derivatives of gM is bounded as well.”
Some problems on mapping class groups and moduli space (2006, background): “,[LSY]), or isoperimetric type versions such a Kobayashi or Kahler hyperbolicity (see [Mc2]).”
The Mapping Class Group from the Viewpoint of Measure Equivalence Theory (2005, unknown): “一方, Gromovの結果 [8]と McMullenの結果 [13]を合わせると, κ(M) ≥ 0となる曲面 M の写像類群 Γ(M) の ℓ2-Betti 数が次のようになることがわかる: βκ(M)+1(Γ(M)) > 0 かつ, 任意の n ̸= κ(M) + 1 に対し, βn(Γ(M)) = 0.”
An analytic construction of the Deligne-Mumford compactification of the moduli space of curves (2013, background): “Petersson in [41] and [42], and there is an extensive amount of literature about it: [17], [18], [26], [43], [40], [46], [47], [49], [51], [52], [53], and [54].”
Canonical Metrics on the Moduli Space of Riemann Surfaces II (2004, methodology): “After we proved this theorem, the second author was informed by McMullen that the equivalence of the Carathéodory metric and the Kobayashi metric may be already known.”