Geodesic-length functions and the weil-petersson curvature tensor

Scott A. Wolpert University of Maryland

Differential Geometry mathscidoc:1609.10273

Journal of Differential Geometry, 91, (2), 321-359, 2012
An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm¨uller and moduli spaces. The tensor is evaluated on the gradients of geodesiclengths 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 Teichm¨uller 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 Teichm¨uller 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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@inproceedings{scott2012geodesic-length,
  title={GEODESIC-LENGTH FUNCTIONS AND THE WEIL-PETERSSON CURVATURE TENSOR},
  author={Scott A. Wolpert},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913230511819028935},
  booktitle={Journal of Differential Geometry},
  volume={91},
  number={2},
  pages={321-359},
  year={2012},
}
Scott A. Wolpert. GEODESIC-LENGTH FUNCTIONS AND THE WEIL-PETERSSON CURVATURE TENSOR. 2012. Vol. 91. In Journal of Differential Geometry. pp.321-359. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913230511819028935.
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