Growth of the Weil-Petersson inradius of moduli space

TL;DR

This paper investigates the growth of the Weil-Petersson inradius of moduli space, showing its relation to genus and punctures, and analyzes the asymptotic behavior of Weil-Petersson volumes of geodesic balls as genus increases.

Contribution

It establishes the asymptotic growth rates of the Weil-Petersson inradius and volumes in moduli space, providing new quantitative insights into their behavior.

Findings

Weil-Petersson inradius grows like √ln(g) with genus g.

Inradius remains comparable to 1 with respect to punctures n.

Volumes of geodesic balls decay faster than any exponential in g.

Abstract

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$. Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of the Weil-Petersson volumes of geodesic balls of finite radii in Teichm\"uller space. We show that they behave like $o((\frac{1}{g})^{(3-\epsilon)g})$ as $g\to \infty$, where $\epsilon>0$ is arbitrary.