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 X1 and Y1, respectively, and that the set X1 −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 Teichmüller space of X onto the Teichmüller space of Y is induced by some quasiconformal map of X onto Y . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.