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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Conformally invariant metrics and prime ends
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by Carl David Minda
Proc. Amer. Math. Soc. 44 (1974), 315-317
DOI: https://doi.org/10.1090/S0002-9939-1974-0338379-6

Abstract:

Let $R$ be a Riemann surface such that the group of conformal self-mappings of $R$ acts transitively on $R$. If $d$ is a metric on $R$ which is invariant under all conformal automorphisms of $R$ and which induces the given topology on $R$, then it is shown that the metric space $\left \langle {R,d} \right \rangle$ is complete. This result is used to show that the prime end compactification of a simply connected Riemann surface $R$ cannot be obtained by completion of a metric space $\left \langle {R,d} \right \rangle$, where $d$ defines the given topology on $R$ and is conformally invariant.
References
  • István S. Gál, Conformally invariant metrics on Riemann surfaces, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1629–1633. MR 112965, DOI 10.1073/pnas.45.11.1629
  • I. S. Gál, Conformally invariant metrics and uniform structures. I. II, Indag. Math. 22 (1960), 218–231, 232–244. Nederl. Akad. Wetensch. Proc. Ser. A 63. MR 0118828
  • Adolf Hurwitz, Vorlesungen über allgemeine Funktionentheorie und elliptische Funktionen, Die Grundlehren der mathematischen Wissenschaften, Band 3, Springer-Verlag, Berlin-New York, 1964 (German). Herausgegeben und ergänzt durch einen Abschnitt über geometrische Funktionentheorie von R. Courant. Mit einem Anhang von H. Röhrl; Vierte vermehrte und verbesserte Auflage. MR 0173749
  • S. Mazurkiewicz, Über die Definition der Primenden, Fund. Math. 26 (1936), 272-279. M. Ohtsuka, Dirichlet problem, extremal length and prime ends, Van Nostrand Reinhold Math. Studies, no. 22, Van Nostrand Reinhold, New York, 1970.
  • Ernest C. Schlesinger, Conformal invariants and prime ends, Amer. J. Math. 80 (1958), 83–102. MR 95269, DOI 10.2307/2372822
  • George Springer, Introduction to Riemann surfaces, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1957. MR 0092855
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Bibliographic Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 44 (1974), 315-317
  • MSC: Primary 30A72
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0338379-6
  • MathSciNet review: 0338379