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)



The Picard theorem for Riemann surfaces

Author: H. L. Royden
Journal: Proc. Amer. Math. Soc. 90 (1984), 571-574
MSC: Primary 30F35; Secondary 30F99
MathSciNet review: 733408
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ W$ be a Riemann surface other than the sphere, plane, punctured plane or torus. Let $ f$ be a holomorphic map of the punctured disk $ 0 < \left\vert z \right\vert < 1$ into $ W$. Then $ f$ can be extended to a holomorphic map of the disk $ \left\vert z \right\vert < 1$, possibly, into a Riemann surface $ {W^ * }$ containing $ W$. We give a new proof of this fact and explore some consequences of it.

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

  • [1] Maurice Heins, On Fuchsoid groups that contain parabolic transformations, Contributions to function theory (Internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 203–210. MR 0151613
  • [2] Heinz Huber, Über analytische Abbildungen Riemannscher Flächen in sich, Comment. Math. Helv. 27 (1953), 1–73 (German). MR 0054051
  • [3] A. Marden, I. Richards, and B. Rodin, Analytic self-mappings of Riemann surfaces, J. Analyse Math. 18 (1967), 197–225. MR 0212182
  • [4] Makoto Ohtsuka, On the behavior of an analytic function about an isolated boundary point, Nagoya Math. J. 4 (1952), 103–108. MR 0048586
  • [5] Makoto Ohtsuka, Boundary components of abstract Riemann surfaces, Lecutres on functions of a complex variable, The University of Michigan Press, Ann Arbor, 1955, pp. 303–307. MR 0069277

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30F35, 30F99

Retrieve articles in all journals with MSC: 30F35, 30F99

Additional Information

Article copyright: © Copyright 1984 American Mathematical Society