Local and Global Aspects of Weil-Petersson Geometry

TL;DR

This survey explores the Weil-Petersson geometry of Teichmuller spaces, emphasizing its negative curvature and convexity properties from differential geometric and CAT(0) perspectives.

Contribution

It provides a comprehensive review of Weil-Petersson geometry focusing on its curvature, convexity, and geometric structures from a differential geometric viewpoint.

Findings

Weil-Petersson metric has negative sectional curvature.

The distance function is convex due to curvature negativity.

Connections to CAT(0) geometry and Coxeter theory are discussed.

Abstract

This is a survey paper on the topic of Weil-Petersson geometry of Teichmuller spaces. Even though historically the subject has been developed as a branch of complex analysis, the treatment here is from the view-point of differential geometry, much influenced by the works of Eells, Earle, Fischer, Tromba and Wolpert over the several decades. The Weil-Petersson geometry has negative sectional curvature, and the main theme of this article is to exploit the convexity of the Weil-Petersson distance function induced by the negativity of the curvature, from Riemannian geometric approaches as well as those of the CAT(0) geometry and Coxeter theory.