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

TL;DR

This paper proves that the Riemannian sectional curvature operator of Teichmüller space with the Weil-Petersson metric is non-positive definite, and shows that certain harmonic maps into this space are constant.

Contribution

It establishes the non-positivity of the curvature operator and applies this to prove harmonic maps from specific rank-one hyperbolic spaces are constant.

Findings

Riemannian sectional curvature operator of Teich(S) is non-positive definite

Any twist harmonic map from certain hyperbolic spaces into Teich(S) is constant

Provides new insights into the geometry of Teichmüller space

Abstract

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 $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_{4}^{-20}/SO(9)$ into $Teich(S)$ is a constant.