Equivalence of the classical theorems of Schottky, Landau, Picard and hyperbolicity
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- by Kyong T. Hahn PDF
- Proc. Amer. Math. Soc. 89 (1983), 628-632 Request permission
Abstract:
Modifying the classical theorems of Schottky and Landau, the author obtains the converses of these theorems. More precisely, the author defines the notions of Schottky, Landau and Picard properties and proves that a plane domain $D$ satisfies any of these properties if and only if ${\mathbf {C}}\backslash D$ contains at least two points. The method of proofs is completely elementary and uses only some basic properties of the Kobayashi metric.References
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D. Bridges, A. Calder and W. Julian, Picard’s theorem, Trans. Amer. Math. Soc. 209 (1982), 513-530.
- Kyong T. Hahn and Kang T. Kim, Hyperbolicity of a complex manifold and other equivalent properties, Proc. Amer. Math. Soc. 91 (1984), no. 1, 49–53. MR 735562, DOI 10.1090/S0002-9939-1984-0735562-9
- Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
- H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin, 1971, pp. 125–137. MR 0304694
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 89 (1983), 628-632
- MSC: Primary 30F10; Secondary 30C99, 32H15
- DOI: https://doi.org/10.1090/S0002-9939-1983-0718986-4
- MathSciNet review: 718986