This is a survey on the analytic properties of the Teichmüller space T(M) of a Riemann surface M, with particular attention to the case where the dimension of T(M) is infinite. Some important results described are due to the authors of this chapter.
The main results presented are the following.
The cotangent space of T(M) is described via the Bers embedding, a biholomorphic embedding of T(M) onto a subset of the Banach space Q(M) of holomorphic quadratic differentials on M. The tangent space is then identified with the Bergman space A1(M), the pre-dual of Q(M). If the surface M is of infinite analytic type, then A1(M) ≠ Q(M), unlike the situation where M is of finite type. The classification of biholomorphic maps between Teichmüller spaces is studied via the classification of the isometries between the cotangent spaces of the corresponding Riemann surfaces, with respect to the Kobayashi and Teichmüller metrics. If M and N are two Riemann surfaces (which a priori need not be homeomorphic), the existence of a surjective ℂ-linear isometry between A1(M) and A1(N) implies that M and N are conformally equivalent. This implies that every biholomorphic map between Teichmüller spaces T(M) and T(N) is induced by a quasiconformal mapping between the surfaces M and N. In particular, the automorphism group of T(M) is equal to the mapping class group of M. This is a complete generalization of a famous theorem of Royden stating the same result for surfaces of finite type. The Teichmüller spaces of any two Riemann surfaces of infinite analytic type are locally bi-Lipschitz equivalent.
The chapter ends with open problems on the subject discussed.