## Journal of Differential Geometry

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

Yunhui Wu

#### 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.

#### Article information

Source
J. Differential Geom., Volume 96, Number 3 (2014), 507-530.

Dates
First available in Project Euclid: 20 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1395321848

Digital Object Identifier
doi:10.4310/jdg/1395321848

Mathematical Reviews number (MathSciNet)
MR3189463

Zentralblatt MATH identifier
1295.30103

#### Citation

Wu, Yunhui. The Riemannian sectional curvature operator of the Weil-Petersson metric and its application. J. Differential Geom. 96 (2014), no. 3, 507--530. doi:10.4310/jdg/1395321848. https://projecteuclid.org/euclid.jdg/1395321848