IMMERSIONS AND EMBEDDINGS IN DOMAINS OF HOLOMORPHY

Let D-1 be a bounded smooth strongly pseudoconvex domain in C-N and le t D-2 be a domain of holomorphy in C-M (2 less than or equal to N, 5 l ess than or equal to M, 2N less than or equal to M). There exists then a proper holomorphic immersion from D-1 to D-2. Furthermore if PI(D-1 , D-2) is the set of proper holomorphic immersions from D-1 to D-2 and A(D-1, D-2) is the set of holomorphic maps from D-1 to D-2 that are c ontinuous on the boundary, then the closure of PI(DI, D-2) in the topo logy of uniform convergence on compacta contains A(D-1, D-2). The appr oximating proper maps can be made tangent to any finite order of conta ct at a given point. The same result was obtained for proper holomorph ic maps, in one codimension, when the target domain has a plurisubharm onic exhaustion function with no saddle critical points. This includes the case where the target domain is convex. Density in a weaker sense was derived in one codimension when the critical points are contained in a compact subset of the target domain. This occurs (for example) w hen the target domain is bounded weakly pseudoconvex with C-2-smooth b oundary. If the target domain is strongly pseudoconvex then the approx imating proper holomorphic maps can also be made continuous on the bou ndary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in C- L is called ''M-dimensional-pseudoconvex'' (where L greater than or eq ual to M) if it has a smooth exhaustion function r such that every poi nt omega in this domain has some M-dimensional complex affine subspace going through this point for which r, restricted to this subspace, is strictly plurisubharmonic in omega. In the result mentioned above the assumption that the target domain is pseudoconvex in C-M (M greater t han or equal to 2N, 5) can be substituted for the assumption that the domain is ''M-dimensional-pseudoconvex''. Similarly, the assumption th at the target domain D-2 is ''(N + 1)-dimensional-pseudoconvex'' and a ll the critical points of some appropriate exhaustion function are ''( N + 1)-dimensional-convex'' (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from D-1 to D-2 cont ains A(D-1, D-2). All the results are obtained with embeddings when th e Euclidean dimensions are such that dim(C)(D-2) greater than or equal to 2dim(C)(D-1) + 1. Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embedding s from D-1 to D-2 contains A(D-1, D-2).