On the Finsler structure of Teichmüller’s metric and Thurston’s metric

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There are several important metrics on Teichmüller space. One of them is the classical Teichmüller metric, introduced by Teichmüller in 1939, and another one is Thurston’s asymmetric metric, introduced by Thurston in 1986. Thurston defined his asymmetric metric in analogy with Teichmüller’s metric, as a solution to an extremal problem. Whereas the Teichmüller distance between two points is defined as the result of searching for the best quasiconformal constant for a map between two conformal structures in a given homotopy class, the Thurston distance is defined as the result of searching for the best Lipschitz constant of a homeomorphism between two hyperbolic structures in a given homotopy class. It turns out that the ideas underlying some known properties of Thurston’s asymmetric metric can be used to get new insight into Teichmüller’s metric, and vice versa. The aim of this paper is to highlight several analogies between the Finsler (infinitesimal) properties of these two metrics. In this direction, in analogy with Thurston’s formula for the Finsler norm of a vector with respect to the asymmetric metric, we give a new formula for the Finsler norm of a vector for Teichmüller’s metric. Whereas the former uses the hyperbolic length function, the latter uses the extremal length function. We also describe an embedding of projective measured foliation space in the cotangent space to Teichmüller space. The image of this embedding is the boundary of the dual of the unit ball for the Finsler structure associated to Teichmüller’s metric.