Teichmüller Spaces and Their Various Metrics

This chapter is a survey on three different kinds of Finsler metrics on Teichmüller spaces. We start from the most classical one, which appeared indeed in the original definition of Teichmüller space, the Teichmüller metric. We give its definition and look at its relation with quadratic differentials and extremal lengths. Then we turn to a hyperbolic analogue of the Teichmüller metric, Thurston’s asymmetric metric, and shall give descriptions of its Finsler structure and of cotangent spaces of Teichmüller space with respect to the norm induced by this metric. In the final part, we turn to the earthquake metric, which was first introduced by Thurston, but has not been seriously studied until quite recently. We shall look at the duality between tangent spaces to Teichmüller space equipped with the earthquake norm and cotangent spaces equipped with Thurston’s norm. We also touch upon the incompleteness of this last metric and its completion.