Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Compact manifolds in hyperbolicity


Author: Robert Brody
Journal: Trans. Amer. Math. Soc. 235 (1978), 213-219
MSC: Primary 32H20
DOI: https://doi.org/10.1090/S0002-9947-1978-0470252-3
MathSciNet review: 0470252
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we establish the strongest possible criterion for the hyperbolicity of a compact complex manifold: such a manifold is hyperbolic if and only if it contains no (nontrivial) complex lines. In addition, we study the behavior of such manifolds under deformation and, in particular, answer the two most natural questions about such deformations: Is the space of hyperbolic complex structures on a given $ {C^\infty }$ manifold open in the space of all its complex structures? (Yes.) Is it closed? (Not in general.) These results answer questions first posed by Kobayashi in [4] and [5].


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

  • [1] Salomon Bochner and Deane Montgomery, Groups on analytic manifolds, Ann. of Math. (2) 48 (1947), 659–669. MR 0022223, https://doi.org/10.2307/1969133
  • [2] A. Douady, Le problème des modules pour les variétés analytiques complexes (d'après M. Kuranishi), Séminaire Bourbaki: 1964/65, Exposé 277, Benjamin, New York and Amsterdam, 1966. MR 33 #54201.
  • [3] Lars Hörmander, An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1966. MR 0203075
  • [4] Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
  • [5] -, Some problems in intrinsic distances and measures, The Greek Math. Soc., C. Carathéodory Symposium, (September, 1973), pp. 306-317.
  • [6] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Springer, Berlin, 1971, pp. 125–137. Lecture Notes in Math., Vol. 185. MR 0304694

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32H20

Retrieve articles in all journals with MSC: 32H20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1978-0470252-3
Article copyright: © Copyright 1978 American Mathematical Society