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 Weil–Petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to − 1 , independently of the genus. This provides a counterexample to some suggestions that the Weil–Petersson metric becomes asymptotically flat, as the genus g goes to infinity, in the thick loci in Teichmüller space. Adopting a different perspective on curvature, we also 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.