The horocyclic metric on Teichm{\"u}ller spaces

TL;DR

This paper explores the earthquake metric on Teichmüller spaces from a conformal perspective, extending Thurston's theory to Riemann surfaces with marked points and establishing a duality similar to Wolpert's in the Finsler setting.

Contribution

It adapts Thurston's earthquake norm to a conformal framework for Riemann surfaces with marked points, introducing a complex Legendre transform and duality analogous to Wolpert's.

Findings

Established a conformal analogue of Thurston's earthquake norm.

Developed a complex Legendre transform for Finsler manifolds.

Proved a Wolpert-like duality in the conformal setting.

Abstract

In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichm{\"u}ller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the surface. This induces a Finsler metric on the Teichm{\"u}ller space, called the earthquake metric. This theory was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the present paper, we study this metric from the conformal viewpoint and we adapt Thurston's theory to the case of Riemann surfaces of arbitrary genus with marked points. A complex version of the Legendre transform defined for Finsler manifolds gives an analogue of the Wolpert duality for the Weil-Petersson symplectic form, which establishes a complete analogue of Thurston's theory of the earthquake norm in the conformal setting. This paper will appear in the Annales de l'Institut Fourier