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

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 t...

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Bibliographic Details
Published in:Journal of differential geometry 2014-03, Vol.96 (3), p.507-530
Main Author: Wu, Yunhui
Format: Article
Language:English
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Summary: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.
ISSN:0022-040X
1945-743X
DOI:10.4310/jdg/1395321848