Uniform Bounds for Weil-Petersson Curvatures

TL;DR

This paper establishes uniform bounds for Weil-Petersson curvatures on Teichmüller space that are independent of surface topology, providing explicit bounds in thick parts and for hyperbolic surfaces.

Contribution

It introduces topology-independent bounds for Weil-Petersson curvatures, including explicit estimates in thick regions and for hyperbolic surfaces.

Findings

Minimal eigenvalue of curvature operator is uniformly bounded away from zero.

Explicit lower bounds for curvature in thick parts of Teichmüller space.

Holomorphic sectional curvature is comparable to -1 in thick hyperbolic surfaces.

Abstract

We find bounds for Weil-Petersson holomorphic sectional curvature, and the Weil-Petersson curvature operator in several regimes, that do not depend on the topology of the underlying surface. Among other results, we show that the minimal (most negative) eigenvalue of the curvature operator at any point in the Teichm\"uller space $\Teich(S_g)$ of a closed surface $S_g$ of genus $g$ is uniformly bounded away from zero. Restricting to a thick part of $\Teich(S_g)$, we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius. We also show that the minimal Weil-Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to $-1$.