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We highlight several analogies between the Finsler (infinitesimal) propertiesof Teichm\"uller's metric and Thurston's asymmetric metric on Teichm\"ullerspace. Thurston defined his asymmetric metric in analogy with Teichm\"ullers'metric, as a solution to an extremal problem, which consists, in the case ofthe asymmetric metric, of finding the best Lipschitz maps in the hoomotopyclass of homeomorphisms between two hyperbolic surface. (In the Teichm\"ullermetric case, one searches for the best quasiconformal map between two conformalsurfaces.) It turns out also that some properties of Thurston's asymmetricmetric can be used to get new insight into Teichm\"uller's metric. In thisdirection, in analogy with Thurston's formula for the Finsler norm of a vectorfor the asymmetric metric that uses the hyperbolic length function, we give anew formula for the Finsler norm of a vector for the Teichm\"uller metric thatuses the extremal length function. We also describe an embedding of projectivemeasured foliation space in the cotangent space to Teichm\"uller space whoseimage is the boundary of the dual of the unit ball in the tangent spacerepresenting vectors of norm one for the Finsler structure associated to theTeichm\"uller metric.

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