The Weil--Petersson geometry of the moduli space of Riemann surfaces

TL;DR

This paper investigates the curvature bounds of the Weil--Petersson metric on moduli spaces of Riemann surfaces, providing new proofs and extending results to Ricci curvature and universal Teichmüller space.

Contribution

It offers a novel proof for curvature bounds in the thick part of the moduli space and extends these bounds to Ricci curvature and the universal Teichmüller space.

Findings

Sectional curvature of Weil--Petersson metric is bounded below by a genus-independent constant in the thick part.

Ricci curvature shares similar lower bounds as sectional curvature.

Universal Teichmüller space has a sectional curvature bounded below by a universal constant.

Abstract

In [4], Z. Huang showed that in the thick part of the moduli space $\mathcal{M}_g$ of compact Riemann surfaces of genus $g$, the sectional curvature of the Weil--Petersson metric is bounded below by a constant depending on injectivity radius, but independent of the genus $g$. In this article, we prove this result by a different method. We also show that the same result holds for Ricci curvature. For the universal Teichm\"uller space equipped with Hilbert structure induced by Weil--Petersson metric, we prove that its sectional curvature is bounded below by a universal constant.