On the Teichm{\"u}ller space of acute triangles

TL;DR

This paper explores the metric structure of the Teichmüller space of acute Euclidean triangles, providing explicit formulas, analyzing rigidity, and describing maximal stretching loci for certain Lipschitz maps.

Contribution

It offers explicit expressions for the distance and Finsler structure on the space of acute triangles, advancing understanding of its geometric properties.

Findings

Explicit formulas for the distance function and Finsler structure.

Result on the infinitesimal rigidity of the metric.

Description of maximal stretching loci for extreme Lipschitz maps.

Abstract

We continue the study of the analogue of Thurston's metric on the Teichm{\"u}ller space of Euclidean triangle which was started by Saglam and Papadopoulos in [1].By direct calculation, we give explicit expressions of the distance function and the Finsler structure of the metric restricted to the subspace of acute triangles.We deduce from the form of the Finsler unit sphere a result on the infinitesimal rigidity of the metric.We give a description of the maximal stretching loci for a family of extreme Lipschitz maps.