On the Finsler stucture of the Teichm\"uuller metric and Thurston's asymmetric metric

Abstract

We highlight several analogies between the Finsler (infinitesimal) properties of Teichm\"uller's metric and Thurston's asymmetric metric on Teichm\"uller space. Thurston defined his asymmetric metric in analogy with Teichm\"ullers' 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 Teichm\"uller 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 Teichm\"uller'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 Teichm\"uller metric that uses the extremal length function. We also describe an embedding of projective measured foliation space in the cotangent space to Teichm\"uller 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 Teichm\"uller metric.