Local rigidity of infinite-dimensional Teichmuller spaces

This paper presents a rigidity theorem for infinite-dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space A(1) (M), for such a Riemann surface M, is isomorphic to the Banach space of summable sequence, l(1). This implies that whenever M and N are Riemann surfaces that are not analytically finite, and in particular are not necessarily homeomorphic, then A(1) (M) is isomorphic to A(1) (N). It is known from V. Markovic that if there is a linear isometry between A(1) (M) and A(1) (N), for two Riemann surfaces M and N of nonexceptional type, then this isometry is induced by a conformal mapping between M and N. As a corollary to this rigidity theorem presented here, taking the Banach duals of A(1) (M) and l(1) shows that the space of holomorphic quadratic differentials on M, Q(M), is isomorphic to the Banach space of bounded sequences, l(infinity). As a consequence of this theorem and the Bers embedding, the Teichmuller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent.