The Riemannian sectional curvature operator of the Weil-Petersson metric and its application

Abstract

Fix a number $g \gt 1$, let $S$ be a close surface of genus $g$, and let $\mathrm{Teich}(S)$ be the Teichmüller space of $S$ endowed with the Weil-Petersson metric. In this paper we show that the Riemannian sectional curvature operator of $\mathrm{Teich}(S)$ 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 $\mathrm{Teich}(S)$ is a constant.