The infinitesimal and global Thurston geometry of Teichm{\"u}ller space

TL;DR

This paper explores the detailed local and global geometric properties of the Thurston metric on Teichmüller space, revealing how tangent and cotangent structures encode surface markings and establishing rigidity results similar to Royden's theorem.

Contribution

It provides a systematic analysis of the infinitesimal geometry of the Thurston metric and derives global rigidity theorems for Teichmüller space.

Findings

Topology, convex, and metric geometry of tangent and cotangent spheres recover surface markings.

Infinitesimal structures lead to global geometric statements.

Rigidity theorems for the Thurston metric analogous to Royden's theorem.

Abstract

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichm{\"u}ller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.