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

TL;DR

This paper investigates the asymptotic behavior of Weil-Petersson curvatures on the moduli space of Riemann surfaces as the genus grows large, revealing new curvature properties in this limit.

Contribution

It provides the first detailed analysis of Weil-Petersson curvature behavior for large genus surfaces, including asymptotics and probabilistic curvature properties.

Findings

Asymptotic behavior of extremal Weil-Petersson holomorphic sectional curvatures

Curvature properties of the moduli space as genus tends to infinity

Probabilistic results on curvature characteristics in large genus limit

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.