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

Fix a number $g>1$, let $S$ be a close surface of genus $g$, and $Teich(S)$ be the Teichm"uller 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 $H_{Q,m}=Sp(m,1)\mathrm{/}Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_4^{-20}\mathrm{/}SO(9)$ into $Teich(S)$ is a constant.