On the Teichmüller space of acute triangles

We continue the study of the analogue of Thurston’s metric on the Teichmüller space of Euclidean triangles which was started by Saglam–Papadopoulos (Minimal stretch maps between Euclidean triangles, 2022). 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.