On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus

Abstract

Let $S_g$ be a closed surface of genus $g$ and ${\mathbb{M}}_g$ be the moduli space of $S_g$ endowed with the Weil-Petersson metric. In this paper we investigate the Weil-Petersson curvatures of ${\mathbb{M}}_g$ for large genus $g$. First, we study the asymptotic behavior of the extremal Weil-Petersson holomorphic sectional curvatures at certain thick surfaces in ${\mathbb{M}}_g$ as $g\to \infty$. Then we prove two curvature properties on the whole space ${\mathbb{M}}_g$ as $g\to \infty$ in a probabilistic way.

30 pages. Comments are welcome!

References in corpus (4)

• The Riemannian Sectional Curvature Operator Of The Weil-Petersson Metric and Its Application
• Understanding Weil-Petersson curvature
• Curvatures of moduli space of curves and applications
• Uniform Bounds for Weil-Petersson Curvatures

Cited by in corpus (1)

• Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures