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On the Finsler stucture of the Teichmüuller metric and Thurston's asymmetric metric

We highlight several analogies between the Finsler (infinitesimal) properties of Teichmüller's metric and Thurston's asymmetric metric on Teichmüller space. Thurston defined his asymmetric metric in analogy with Teichmüllers' metric, as a solution to an extremal problem, which consists, in the case of the asymmetric metric, of finding the best Lipschitz maps in the hoomotopy class of homeomorphisms between two hyperbolic surface. (In the Teichmüller metric case, one searches for the best quasiconformal map between two conformal surfaces.) It turns out also that some properties of Thurston's asymmetric metric can be used to get new insight into Teichmüller's metric. In this direction, in analogy with Thurston's formula for the Finsler norm of a vector for the asymmetric metric that uses the hyperbolic length function, we give a new formula for the Finsler norm of a vector for the Teichmüller metric that uses the extremal length function. We also describe an embedding of projective measured foliation space in the cotangent space to Teichmüller space whose image is the boundary of the dual of the unit ball in the tangent space representing vectors of norm one for the Finsler structure associated to the Teichmüller metric.