Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric

Hideki Miyachi (1) , Ken'Ichi Ohshika (2) , Athanase Papadopoulos (3)

1 Kanazawa University, School of Mathematics and Physics
2 Department of Mathematics [Gakushuin]
3 IRMA - Institut de Recherche Mathématique Avancée

In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichmüller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space to Teichmüller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichmüller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.

Mots clés: Thurston metric; Teichmüller space; Teichmüller metric; Finsler manifold

Domaines: Topologie géométrique [math.GT]; Géométrie métrique [math.MG]; Variables complexes [math.CV]

Hideki Miyachi, Ken'Ichi Ohshika, Athanase Papadopoulos. Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric. 2020. ⟨hal-02570185v1⟩