Uniform Bounds for Weil-Petersson Curvatures

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 Teichmüller 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 $-1$.

To appear in Proceedings of the London Mathematical Society, 41 pages

References in corpus (1)

• The Riemannian Sectional Curvature Operator Of The Weil-Petersson Metric and Its Application

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• On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus
• Estimates for the average scalar curvature of the Weil-Petersson metric on the Moduli space ${\overline{\mathcal{M}}}_g$