The notions of hyperbolicity and tautness modulo an analytic subset of complex spaces are due to S. Kobayashi. Much attention has been given to these notions, and the results on this problem can be applied to many areas of mathematics, in particular to the extensions of holomorphic mappings. The main goal of this article is to give necessary and sufficient conditions on the hyperbolicity or tautness modulo an analytic subset of complex spaces. | JOURNAL OF SCIENCE OF HNUE Mathematical and Physical Sci. 2014 Vol. 59 No. 7 pp. 34-43 This paper is available online at http ON HYPERBOLICITY AND TAUTNESS MODULO AN ANALYTIC SUBSET OF COMPLEX SPACES Mai Anh Duc Faculty of Mathematics - Physics and Informatics Tay Bac University Abstract. The main goal of this article is to give necessary and sufficient conditions on the hyperbolicity or tautness modulo an analytic subset of complex spaces. Keywords Hyperbolicity modulo analytic subset tautness modulo. 1. Introduction The notions of hyperbolicity and tautness modulo an analytic subset of complex spaces are due to S. Kobayashi see 2 p. 68 . Much attention has been given to these notions 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 2 and 3 . The main goal of this article is to give necessary and sufficient conditions on the hyperbolicity or tautness modulo an analytic subset of complex spaces. 2. Basic Notions First of all we recall the definitions of hyperbolicity and tautness modulo an analytic subset of complex spaces. Definition . see 3 p. 68 Let X be a complex space and S be an analytic subset of X. We say that X is hyperbolic modulo S if for every pair of distinct points p q of X we have dX p q gt 0 unless both are contained in S where dX is the Kobayashi pseudodistance of X. If S then X is said to be hyperbolic. Received July 28 2014. Accepted August 30 2014. Contact Mai Anh Duc e-mail address ducphuongma@ 34 On hyperbolicity and tautness modulo an analytic subset of complex spaces Definition . see 3 Let X be a complex space and S be an analytic subset in X. We say that X is taut modulo S if it is normal modulo S . for every sequence fn in Hol D X one of the following holds i. There exists a subsequence of fn which converges uniformly to f Hol D X in Hol D X ii. The sequence fn is compactly divergent modulo S in Hol
