Geodesic
Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric
Hideki Miyachi1 ; Ken'ichi Ohshika2 ; Athanase Papadopoulos3
1Kanazawa University, College of Science and Engineering, School of Mathematics and Physics
2Gakushuin University, Department of Mathematics
3Université de Strasbourg et CNRS, Institut de Recherche Mathématique Avancée
Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 325-334

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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 circle 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 hyperbolic metric). We describe the unit tangent circles of the metrics in this family.
DOI : 10.54330/afm.113702
Keywords: Thurston metric, Teichmüller space, Teichmüller metric, Finsler manifold

Hideki Miyachi  1   ; Ken'ichi Ohshika  2   ; Athanase Papadopoulos  3

1 Kanazawa University, College of Science and Engineering, School of Mathematics and Physics
2 Gakushuin University, Department of Mathematics
3 Université de Strasbourg et CNRS, Institut de Recherche Mathématique Avancée 
Hideki Miyachi; Ken'ichi Ohshika; Athanase Papadopoulos. Tangent spaces of the Teichmüller space of the torus with Thurston's weak metric. Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 325-334. doi: 10.54330/afm.113702
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