Uniform bounds on harmonic Beltrami differentials and Weil-Petersson curvatures

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.

arXiv commentAny commments are welcome

References in corpus (3)

• The Riemannian Sectional Curvature Operator Of The Weil-Petersson Metric and Its Application
• Uniform Bounds for Weil-Petersson Curvatures
• On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus

Cited by in corpus (4)

• The Weil-Petersson gradient flow of renormalized volume and 3-dimensional convex cores
• A new uniform lower bound on Weil-Petersson distance
• Systole functions and Weil-Petersson geometry
• Estimates for the average scalar curvature of the Weil-Petersson metric on the Moduli space ${\overline{\mathcal{M}}}_g$