Essential regularity of the model space for the Weil–Petersson metric

Abstract

Abstract This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion {\overline{\mathcal{T}}} of Teichmüller space. The boundary of {\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space {\overline{\bf H}} called the model space. The significance of {\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of {\overline{\mathcal{T}}} . The main result of the paper is that certain subsets of {\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on {\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.