Journal of Differential Geometry

Geodesic-length functions and the Weil-Petersson curvature tensor

Scott A. Wolpert

Full-text: Open access

Abstract

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.

Article information

Source
J. Differential Geom., Volume 91, Number 2 (2012), 321-359.

Dates
First available in Project Euclid: 8 August 2012

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1344430826

Digital Object Identifier
doi:10.4310/jdg/1344430826

Mathematical Reviews number (MathSciNet)
MR2971291

Zentralblatt MATH identifier
1252.32022

Citation

Wolpert, Scott A. Geodesic-length functions and the Weil-Petersson curvature tensor. J. Differential Geom. 91 (2012), no. 2, 321--359. doi:10.4310/jdg/1344430826. https://projecteuclid.org/euclid.jdg/1344430826


Export citation