Geodesic-length functions and the Weil-Petersson curvature tensor

TL;DR

This paper develops an expansion for the Weil-Petersson curvature tensor in thin regions of Teichmüller space, providing bounds, continuity properties, and a classification of asymptotic flats relevant to the geometry of moduli spaces.

Contribution

It introduces a detailed expansion of the Weil-Petersson curvature tensor, analyzes its asymptotic product structure, and classifies flats and curvature properties near the boundary of Teichmüller space.

Findings

Lower bounds for sectional curvature in terms of surface systole.

Continuity properties of the curvature tensor at augmented space boundary.

Classification of asymptotic flats and negative curvature in tangent subspaces.

Abstract

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichm\"{u}ller 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 Teichm\"{u}ller 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\"{u}ller space contain sections with a definite amount of negative curvature. Proofs combine estimates for uniformization group exponential-distance sums and potential theory bounds.