Illinois Journal of Mathematics Volume 48, Number 4, Winter 2004, Pages 1367–1384 S 0019-2082 GENERALIZATIONS OF THE THEOREMS OF CARTAN AND GREENE–KRANTZ TO COMPLEX MANIFOLDS DO DUC THAI AND TRAN HUE MINH Abstract. In this article, some generalizations of the theorems of Car- tan and Greene–Krantz for the family of biholomorphic mappings on (not necessary bounded) domains of a complex manifold are given. Moreover, a necessary and sufficient condition for strongly complete C- hyperbolicity of domains in a complex manifold with compact quotients is obtained. 1. Introduction H. Cartan [4] proved the following theorem about compactness of families of biholomorphic mappings (see also [13, Thm. 4, p. 78]). Theorem. Let Ω be a bounded domain in C n . Suppose {f i } is a sequence of biholomorphic mappings f i :Ω → Ω which converges uniformly on com- pact subsets of Ω to a mapping f . Then the following three conditions are equivalent: (i) f is a biholomorphic mapping of Ω onto Ω. (ii) f (Ω) is not a subset of ∂ Ω, the boundary of Ω in C n . (iii) The Jacobian determinant det[f 0 (z)] is not identically zero on Ω. Much attention has been given to generalizations of Cartan’s theorem. For instance, under some additional hypotheses on the domains and the mappings, S. Bell [3] and W. Klingenberg and S. Pinchuk [10] proved the above theorem with “biholomorphic” replaced by “proper”. However, as far as we know, the problem of generalizing Cartan’s theorem to unbounded domains in a complex manifold remains open. The first purpose of this paper is to give several versions of Cartan’s the- orem for families of biholomorphic mappings on a (not necessary bounded) domain of a complex manifold from the viewpoint of hyperbolic complex anal- ysis. Namely, we will prove the following results: Received April 17, 2004; received in final form May 10, 2004. 2000 Mathematics Subject Classification. Primary 32H02, 32D20. Secondary 32H20, 32F30. c 2004 University of Illinois 1367