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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

On moduli of plane domains


Author: Ignacio Guerrero
Journal: Proc. Amer. Math. Soc. 67 (1977), 41-49
MSC: Primary 32G15; Secondary 30A46
MathSciNet review: 0454074
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Abstract: It is well known that an arbitrary plane domain of finite connectivity can be mapped conformally onto an annulus minus a certain number of circular slits. The parameters defining such a canonical domain are studied in the context of Teichmüller theory.

Let $ \Omega $ be a plane domain bounded by $ m \geqslant 3$ continua. Denote by $ T(\Omega )$ the reduced Teichmüller space of $ \Omega $ and by $ R(\Omega )$ the space of conformal equivalence classes of domains bounded, as $ \Omega $ is, by m continua. A real analytic map from $ T(\Omega )$ onto an open subset $ S(\Omega )$ of a $ 3m - 6$ dimensional product of circles and lines is constructed. It is shown that the map $ T(\Omega ) \to S(\Omega )$ is a regular covering map. Finally, it is observed that there is a finite sheeted covering map $ S(\Omega ) \to R(\Omega )$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0454074-X
PII: S 0002-9939(1977)0454074-X
Keywords: Plane domain, moduli, Teichmüller space
Article copyright: © Copyright 1977 American Mathematical Society