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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. 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

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

Official Date:

August 2006

Dates:

Date	Event
August 2006	Published

Volume:

Vol.74

Number:

No.1

Number of Pages:

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

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