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In this paper, we show that the analogue of Thurston's asymmetric metric onthe Teichm{\"u}ller space of flat structures on the torus is weak Finsler andwe give a geometric description of its unit sphere at each point in the tangentspace to Teichm{\"u}ller space. We then introduce a family of weak Finslermetrics which interpolate between Thurston's asymmetric metric and theTeichm{\"u}ller metric of the torus (which coincides with the the hyperbolicmetric). We describe the infinitesimal unit spheres of the metrics in thisfamily.The final version of this paper will appear in Annales Academi\ae \Scientiarum Fennic\ae \ Mathematica.

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