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Local rigidity of infinite-dimensional Teichmüller spaces

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Fletcher, A. (Alastair) (2006) Local rigidity of infinite-dimensional Teichmüller spaces. Journal of the London Mathematical Society, Vol.74 (No.1). pp. 26-40. doi:10.1112/S0024610706023003

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Official URL: http://dx.doi.org/10.1112/S0024610706023003

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > 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: 0024-6107
Official Date: August 2006
Dates:
DateEvent
August 2006Published
Volume: Vol.74
Number: No.1
Number of Pages: 15
Page Range: pp. 26-40
DOI: 10.1112/S0024610706023003
Status: Peer Reviewed
Access rights to Published version: Open Access

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