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 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.