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Assume both X and Y are Riemann surfaces which are subsets of compact Riemann surfaces X1 and Y1; respectively, and that the set X1 X has innitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic dierentials on X and Y are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmuller space of X onto the Teichmuller space of Y is induced by some quasiconformal map of X onto Y. Consequently we can nd an uncountable set of Riemann surfaces whose Teichmuller spaces are not biholomorphically equivalent.

An isometry theorem for quadratic differentials on Riemann surfaces of finite genus