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

Assume both X and Y are Riemann surfaces which are subsets of compact Riemann surfaces X-1 and Y-1, respectively, and that the set X-1 - X has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials 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 find an uncountable set of Riemann surfaces whose Teichmuller spaces are not biholomorphically equivalent.