Local and Global Aspects of Weil-Petersson Geometry

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.