A characterization of hyperbolic manifolds

Abate, Marco

doi:10.1090/S0002-9939-1993-1128723-6

A characterization of hyperbolic manifolds
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Proc. Amer. Math. Soc. 117 (1993), 789-793 Request permission

Abstract:

In this note we prove that a complex manifold $X$ is Kobayashi hyperbolic if and only if the space $\operatorname {Hol} (\Delta ,X)$ of holomorphic maps of the unit disk $\Delta$ into $X$ is relatively compact (with respect to the compact-open topology) in the space $C(\Delta ,{X^{\ast }})$ of continuous maps from $\Delta$ into the one-point compactification ${X^{\ast }}$ of $X$.

References

Marco Abate, Iteration theory of holomorphic maps on taut manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989. MR 1098711
Theodore J. Barth, Taut and tight complex manifolds, Proc. Amer. Math. Soc. 24 (1970), 429–431. MR 252679, DOI 10.1090/S0002-9939-1970-0252679-6
Theodore J. Barth, The Kobayashi distance induces the standard topology, Proc. Amer. Math. Soc. 35 (1972), 439–441. MR 306545, DOI 10.1090/S0002-9939-1972-0306545-X
John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Peter Kiernan, On the relations between taut, tight and hyperbolic manifolds, Bull. Amer. Math. Soc. 76 (1970), 49–51. MR 252678, DOI 10.1090/S0002-9904-1970-12363-1
Shoshichi Kobayashi, Invariant distances on complex manifolds and holomorphic mappings, J. Math. Soc. Japan 19 (1967), 460–480. MR 232411, DOI 10.2969/jmsj/01940460
Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
Shoshichi Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), no. 3, 357–416. MR 414940, DOI 10.1090/S0002-9904-1976-14018-9
Serge Lang, Introduction to complex hyperbolic spaces, Springer-Verlag, New York, 1987. MR 886677, DOI 10.1007/978-1-4757-1945-1
Jean-Pierre Rosay, Un exemple d’ouvert borné de $\textbf {C}^{3}$ “taut” mais non hyperbolique complet, Pacific J. Math. 98 (1982), no. 1, 153–156 (French). MR 644946
H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193–233. MR 224869, DOI 10.1007/BF02392083