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## Local rigidity of infinite-dimensional Teichmüller spaces

### Abstract

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 non-exceptional 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, $l^{\infty }$. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally bi-Lipschitz equivalent

Topics: QA
Publisher: Cambridge University Press
Year: 2006
OAI identifier: oai:wrap.warwick.ac.uk:692

### Citations

1. (2003). Biholomorphic maps between Teichm¨ uller spaces’,
2. (1994). c, ‘The dual of the Bergman space deﬁned on a hyperbolic plane domain’,
3. (1971). Contributions to the theory of the classical Banach spaces’,
4. (1984). Hp-spaces, p 1, and spline systems’,
5. (1960). Projections in certain Banach spaces’,
6. (2000). Quasiconformal Teichm¨ uller theory
7. (1987). Real and complex analysis (McGraw-Hill,
8. (1980). Representation theorems for Hardy spaces’,
9. (1987). Teichm¨ uller theory and quadratic diﬀerentials
10. (1987). Univalent functions and Teichm¨ uller spaces (Springer,

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