Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 )$.

References [Enhancements On Off] (What's this?)

  • [1] L. V. Ahlfors, The complex analytic structure of the space of closed Riemann surfaces, Analytic Functions, Princeton Univ. Press, Princeton, N. J., 1960, pp. 45-66. MR 0124486 (23:A1798)
  • [2] -, Complex analysis, second ed., McGraw-Hill, New York, 1966.
  • [3] -, Lectures on quasiconformal mappings, Van Nostrand, Princeton, N. J., 1966. MR 0200442 (34:336)
  • [4] L. Bers, A nonstandard integral equation with applications to quasiconformal mappings, Acta Math. 116 (1967), 1078-1082. MR 0192046 (33:273)
  • [5] -, Holomorphic differentials as functions of moduli, Bull. Amer. Math. Soc. 67 (1961), 206-210. MR 0122989 (23:A320)
  • [6] C. J. Earle, Teichmüller spaces of groups of the second kind, Acta Math. 112 (1964), 91-97. MR 0165096 (29:2385)
  • [7] -, Reduced Teichmüller spaces, Trans. Amer. Math. Soc. 126 (1967), 54-63. MR 0204642 (34:4481)
  • [8] H. E. Rauch, On the transcendental moduli of algebraic Riemann surfaces, Proc. Nat. Acad. Sci. U.S.A. 40 (1955), 42-49. MR 0072232 (17:251e)
  • [9] M. Schiffer and D. C. Spencer, Functionals of finite Riemann surfaces, Princeton Univ. Press, Princeton, N. J., 1954. MR 0065652 (16:461g)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32G15, 30A46

Retrieve articles in all journals with MSC: 32G15, 30A46

Additional Information

Keywords: Plane domain, moduli, Teichmüller space
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society