A new uniform lower bound on Weil-Petersson distance

TL;DR

This paper establishes a new uniform lower bound on the Weil-Petersson distance by analyzing the Lipschitz continuity of the square root of the injectivity radius and systole function on Teichmüller space, with implications for the geometry of moduli space.

Contribution

It provides the first explicit uniform Lipschitz bounds for the square root of the injectivity radius and systole function in Weil-Petersson geometry, enhancing understanding of Teichmüller space's metric properties.

Findings

Square root of injectivity radius is 0.3884-Lipschitz on Teichmüller space.

Square root of systole function is 0.5492-Lipschitz on Teichmüller space.

Results have applications to the asymptotic geometry of moduli space for large genus.

Abstract

In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $ 0.3884$-Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be chosen to be $0.5492$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.