The Library

Local rigidity of infinite-dimensional Teichmüller spaces

Tools

Fletcher, A. (Alastair). (2006) Local rigidity of infinite-dimensional Teichmüller spaces. Journal of the London Mathematical Society, Vol.74 (No.1). pp. 26-40. ISSN 0024-6107

Preview

PDF
WRAP_Fletcher_Local_rigidity.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (226Kb)

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
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:	0024-6107
Date:	August 2006
Volume:	Vol.74
Number:	No.1
Number of Pages:	15
Page Range:	pp. 26-40
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. 2. F. Gardiner, Teichmüller theory and quadratic differentials (Wiley, New York, 1987) 3. F. Gardiner and N. Lakic, Quasiconformal Teichmüller theory (American Mathematical Society, Providence, RI, 2000). 4. O. Lehto, Univalent functions and Teichmüller spaces (Springer, New York, 1987). 5. A. Pelczynski, ‘Projections in certain Banach spaces’, Studia Math. 19 (1960) 209–228. 6. J. Lindenstrauss and A. Pelczynski, ‘Contributions to the theory of the classical Banach spaces’, J. Funct. Anal. 8 (1971) 225–249. 7. M. Mateljevic, ‘The dual of the Bergman space defined on a hyperbolic plane domain’, Publ. Inst. Math. (Beograd) (N.S.) 56(70) (1994) 135–139. 8. V. Marković ‘Biholomorphic maps between Teichmüller spaces’, Duke Math. J. 120 (2003) 405–431. 9. W. Rudin, Real and complex analysis (McGraw-Hill, New York, 1987). 10. P. Wojtaszczyk, ‘Hp-spaces, p ≤ 1, and spline systems’, Studia Math. 77 (1984) 289–320.
URI:	http://wrap.warwick.ac.uk/id/eprint/692