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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 ISSN 0024-6107 .
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Official URL: http://dx.doi.org/10.1112/S0024610706023003
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 | ||||
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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: |
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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 (Creative Commons) |
Data sourced from Thomson Reuters' Web of Knowledge
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