The Weil-Petersson metric geometry

TL;DR

This paper explores the geometric structure of the Weil-Petersson metric on Teichmüller space, detailing coordinate descriptions, gradient and Hessian formulas, and applications, including tangent cone analysis and comparisons with flat tori.

Contribution

It provides a comprehensive description of Weil-Petersson geometry using Fenchel-Nielsen coordinates and introduces new formulas for geodesic-length functions and tangent cone structures.

Findings

Formulas for gradients and Hessians of geodesic-length functions.

Description of the Weil-Petersson metric in Fenchel-Nielsen coordinates.

Analysis of the Alexandrov tangent cone at augmentation points.

Abstract

A summary introduction of the Weil-Petersson metric space geometry is presented. Teichmueller space and its augmentation are described in terms of Fenchel-Nielsen coordinates. Formulas for the gradients and Hessians of geodesic-length functions are presented. Applications are considered. A description of the Weil-Petersson metric in Fenchel-Nielsen coordinates is presented. The Alexandrov tangent cone at points of the augmentation is described. A comparison dictionary is presented between the geometry of the space of flat tori and Teichmueller space with the Weil-Petersson metric.