Systole functions and Weil-Petersson geometry

Abstract

A basic feature of Teichmüller theory of Riemann surfaces is the interplay of two dimensional hyperbolic geometry, the behavior of geodesic-length functions and Weil-Petersson geometry. Let ${\mathcal{T}}_g$ $(g\ge 2)$ be the Teichmüller 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 $L^p$ $(1\le p\le \infty )$-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.

Mathematische Annalen, to appear, 32 pages, any comments are welcome