Systole functions and Weil-Petersson geometry

TL;DR

This paper investigates the relationship between systolic curves and Weil-Petersson geometry in Teichmüller space, establishing uniform bounds on norms of geodesic-length gradients and the minimal holomorphic sectional curvature.

Contribution

It provides new uniform estimates for geodesic-length gradients and shows the minimal Weil-Petersson holomorphic sectional curvature is bounded above by a negative constant independent of genus.

Findings

L^p norms of gradients are comparable to systole^{1/p}

Minimal Weil-Petersson holomorphic sectional curvature is uniformly bounded above by a negative constant

Answers a question of M. Mirzakhani about curvature bounds

Abstract

A basic feature of Teichm\"uller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let $\mathcal{T}_g$ $(g\geq 2)$ be the Teichm\"uller space of closed Riemann surfaces of genus $g$. Our goal in this paper is to study the gradients of geodesic-length functions along systolic curves. We show that their $L^p$ $(1\leq p \leq \infty)$-norms at every hyperbolic surface $X\in \mathcal{T}_g$ are uniformly comparable to $\ell_{sys}(X)^{\frac{1}{p}}$ where $\ell_{sys}(X)$ is the systole of $X$. As an application, we show that the minimal Weil-Petersson holomorphic sectional curvature at every hyperbolic surface $X\in \mathcal{T}_g$ is bounded above by a uniform negative constant independent of $g$, which negatively answers a question of M. Mirzakhani. Some other applications to the geometry of $\mathcal{T}_g$ will also be discussed.