Surveys in Differential Geometry
Volume 14 (2009)
Geometry of Teichmüller space with the Teichmüller metric
Pages: 295 – 314
The purpose of this chapter is to describe recent progress in the study of Teichmüller geometry. We focus entirely on the Teichmüller metric. A survey of the very important Weil-Petersson metric can be found in [W]. The study of the Teichmüller metric has different aspects. One major theme in the subject is to what extent Teichmüller space with the Teichmüller metric resembles a metric of negative curvature, and to what extent it resembles a metric of nonnegative curvature. This theme will occupy much of this survey and we will describe results in both directions. With somewhat the same theme we describe recent results about the Teichmüller geodesic flow on moduli space.