Biholomorphic maps between Teichmüller spaces

Abstract

In this paper we study biholomorphic maps between Teichmüller spaces and the induced linear isometries between the corresponding tangent spaces. The first main result in this paper is the following classification theorem. If $M$ and $N$ are two Riemann surfaces that are not of exceptional type, and if there exists a biholomorphic map between the corresponding Teichmüller spaces Teich($M$) and Teich($N$), then $M$ and $N$ are quasiconformally related. Also, every such biholomorphic map is geometric. In particular, we have that every automorphism of the Teichmüller space Teich($M$) must be geometric. This result generalizes the previously known results (see [2], [5], [7]) and enables us to prove the well-known conjecture that states that the group of automorphisms of Teich($M$) is isomorphic to the mapping class group of $M$ whenever the surface $M$ is not of exceptional type. In order to prove the above results, we develop a method for studying linear isometries between $L^1$-type spaces. Our focus is on studying linear isometries between Banach spaces of integrable holomorphic quadratic differentials, which are supported on Riemann surfaces. Our main result in this direction (Theorem 1.1) states that if $M$ and $N$ are Riemann surfaces of nonexceptional type, then every linear isometry between $A^1(M)$ and $A^1(N)$ is geometric. That is, every such isometry is induced by a conformal map between $M$ and $N$.