Geometry of the Weil-Petersson completion of Teichm\"uller space

TL;DR

This paper explores the geometric properties of Weil-Petersson geodesics in the completed Teichmüller space, emphasizing CAT(0) geometry, and provides simplified proofs and classifications related to geodesic behavior.

Contribution

It offers a new perspective on WP geometry, including simplified proofs of key theorems and classifications of flats and geodesic limits in the WP CAT(0) setting.

Findings

Simplified proof of the Masur-Wolf theorem

Classification of flats in WP geometry

Analysis of geodesic limits in the WP completion

Abstract

We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichm\"uller space. We sketch a collection of results by other authors and then proceed to develop the properties of the WP CAT(0) geometry. Our approach includes a simplified proof of the Masur-Wolf theorem, a classification of flats and of geodesic limits.