Open Access
June 2012 Geodesic-length functions and the Weil-Petersson curvature tensor
Scott A. Wolpert
J. Differential Geom. 91(2): 321-359 (June 2012). DOI: 10.4310/jdg/1344430826


An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichmüller and moduli spaces. The tensor is evaluated on the gradients of geodesic lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvature tensor expansion is applied to establish continuity properties at the frontier strata of the augmented Teichmüller space. The curvature tensor has the asymptotic product structure already observed for the metric and covariant derivative. The product structure is combined with the earlier negative sectional curvature results to establish a classification of asymptotic flats. Furthermore, tangent subspaces of more than half the dimension of Teichmüller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds.


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Scott A. Wolpert. "Geodesic-length functions and the Weil-Petersson curvature tensor." J. Differential Geom. 91 (2) 321 - 359, June 2012.


Published: June 2012
First available in Project Euclid: 8 August 2012

zbMATH: 1252.32022
MathSciNet: MR2971291
Digital Object Identifier: 10.4310/jdg/1344430826

Rights: Copyright © 2012 Lehigh University

Vol.91 • No. 2 • June 2012
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