# Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures

## TL;DR

This paper establishes uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures on hyperbolic surfaces, revealing their behavior in relation to systole length and genus growth.

## Contribution

It provides new uniform bounds linking harmonic Beltrami differentials and Weil-Petersson curvatures to geometric parameters of hyperbolic surfaces.

## Key findings

- Bound on harmonic Beltrami differentials at points of small injectivity radius.
- Lower bound on Weil-Petersson Ricci curvature in terms of systole.
- Average Weil-Petersson scalar curvature scales with genus g.

## Abstract

In this article we show that for every finite area hyperbolic surface $X$ of type $(g,n)$ and any harmonic Beltrami differential $\mu$ on $X$, then the magnitude of $\mu$ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of $\mu$ over the square root of the systole of $X$ up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil-Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil-Petersson scalar curvature over the moduli space is uniformly comparable to $-g$ as the genus $g$ goes to infinity.