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Local rigidity of infinitedimensional Teichmüller spaces
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Fletcher, A. (Alastair). (2006) Local rigidity of infinitedimensional Teichmüller spaces. Journal of the London Mathematical Society, Vol.74 (No.1). pp. 2640. ISSN 00246107

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Official URL: http://dx.doi.org/10.1112/S0024610706023003
Abstract
This paper presents a rigidity theorem for infinitedimensional 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, $l^{\infty }$. As a consequence of this theorem and the Bers embedding, the Teichmüller spaces of such Riemann surfaces are locally biLipschitz equivalent.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Mathematics 
Library of Congress Subject Headings (LCSH):  Teichmüller spaces, Bergman spaces, Riemann surfaces, Lipschitz spaces, Functions of several complex variables 
Journal or Publication Title:  Journal of the London Mathematical Society 
Publisher:  Cambridge University Press 
ISSN:  00246107 
Official Date:  August 2006 
Volume:  Vol.74 
Number:  No.1 
Number of Pages:  15 
Page Range:  pp. 2640 
Identification Number:  10.1112/S0024610706023003 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  1. R. Coifman and R. Rochberg, ‘Representation theorems for Hardy spaces’, Astèrisque 77 (1980) 11–66. 
URI:  http://wrap.warwick.ac.uk/id/eprint/692 
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