Hyperbolic Imbeddedness and Extensions of Holomorphic Mappings

Abstract

Abstract. In this article, the hyperbolic imbeddedness of (not necessary relatively compact) complex subspaces of a complex space is investigated. Using these results, two theorems on extending holomorphic mappings through hypersurfaces are given. IntroductionCharacterizing hyperbolic imbeddedness in the sense of Kobayashi is one of the most important problems of hyperbolic complex analysis. Much attention has been given to this problem, and the results on this problem can be applied to many areas of mathematics, in particular to the extensions of holomorphic mappings. For details see [La] and [Ko].Recall that a complex subspace M of a complex space X is hyperbolically imbedded in X if for distinct p, q ∈ M, the closure of M, there are open setsUnfortunately, as far as we know, almost all studies of hyperbolic imbeddedness were done under the assumption on the relative compactness of M in X. Motivated by studying geometry of unbounded domains in C n , the investigation of hyperbolic imbeddedness of (not necessary relatively compact) complex subspaces of a complex space is posed. In this paper we focus on such studies and use these results to extend holomorphic mappings through hypersurfaces, hoping that our results will be useful to people working on that subject.We now give a brief outline of the content of this article. In Section 1 we review some basic notions needed later. In Section 2 we study hyperbolic imbeddedness 2000 Mathematics Subject Classification: Primary 32H05, 32H15.