Local rigidity of infinite dimensional Teichmüller Spaces

This paper presents a rigidity theorem for infinite dimensional Bergman spaces of hyperbolic Riemann surfaces, which states that the Bergman space A1(M), for such a Riemann surface M , is isomorphic to the Banach space of summable sequence, l1. This implies that whenever M and N are Riemann surfaces which are not analytically finite, and in particular are not necessarily homeomorphic, then A1(M) is isomorphic to A1(N). It is known from V. Markovic that if there is a linear isometry between A1(M) and A1(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 A1(M) and l1 shows that the space of holomorphic quadratic differentials on M , Q(M), is isomorphic to the Banach space of bounded sequences, l∞. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent. 1. Definitions and Introduction In this paper, M will be a hyperbolic Riemann surface with the unit disk as its universal cover, and Γ is the covering group such thatM ' D/Γ. The Banach space L(M) is the space of measurable functions on M with norm ||φ||1 = ∫ M |φ| <∞. Unless confusion arises, ||φ|| will mean ||φ||1 in this paper. The Bergman space A(M) ⊂ L(M) is the Banach space of holomorphic functions integrable on M . The Bers space Q(M) is the Banach space of holomorphic quadratic differentials on M with norm ||φ||Q = sup z∈M ρ−2 M (z)|φ(z)| <∞, where φ ∈ Q and ρM is the hyperbolic density on M . The space of absolutely summable sequences is l = { (a0, a1, ...) : ai ∈ C, ∞ ∑