Tangent spaces of the Teichm{\"u}ller space of the torus with Thurston's weak metric

Abstract

In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{\"u}ller 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 Teichm{\"u}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichm{\"u}ller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.The final version of this paper will appear in Annales Academi\ae \ Scientiarum Fennic\ae \ Mathematica.