Local and global aspects of Weil–Petersson geometry

Abstract

This is a survey on the topic of Weil–Petersson geometry of Teichmüller spaces. Although 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 the Coxeter theory. Also we discuss the Weil–Petersson geometry of the universal Teichmüller space. In particular, the linear deformation theory of the hyperbolic metrics is treated synthetically in the sense that the universal Teichmüller space and the finite dimensional Teichmüller space are contrasted with each other.