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

Abstract

Fix a number $g>1$, let $S$ be a close surface of genus $g$, and $Teich(S)$ be the Teichmüller space of $S$ endowed with the Weil-Petersson metric. In this paper we show that the Riemannian sectional curvature operator of $Teich(S)$ is non-positive definite. As an application we show that any twist harmonic map from rank-one hyperbolic spaces $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_4^{-20}/SO(9)$ into $Teich(S)$ is a constant.

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• Uniform Bounds for Weil-Petersson Curvatures
• On the Weil-Petersson curvature of the moduli space of Riemann surfaces of large genus
• The Weil-Petersson curvature operator on the universal Teichmüller space
• Harmonic maps from Kähler manifolds
• Estimates for the average scalar curvature of the Weil-Petersson metric on the Moduli space ${\overline{\mathcal{M}}}_g$