The vanishing rate of Weil-Petersson sectional curvatures

TL;DR

This paper investigates how the Weil-Petersson sectional curvatures tend to zero on Riemann surfaces with short geodesics, providing bounds and examples to understand the rate of curvature vanishing.

Contribution

It establishes bounds on Weil-Petersson sectional curvature near the boundary of the moduli space and explores the rate at which these curvatures vanish.

Findings

Bound the sectional curvature away from zero using lengths of short geodesics

Construct examples illustrating curvature behavior near the boundary

Propose an expectation for the actual rate of curvature vanishing

Abstract

The Weil-Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil-Petersson metric is approximately a product metric. An almost product metric has sections with almost vanishing curvature. We bound the sectional curvature away from zero in terms of the product of lengths of short geodesics on Riemann surfaces. We give examples and an expectation for the actual vanishing rate.