Immersions and embeddings in domains of holomorphy

Abstract

Let D 1 {D_1} be a bounded smooth strongly pseudoconvex domain in C N {\mathbb {C}^N} and let D 2 {D_2} be a domain of holomorphy in C M ( 2 ⩽ N , 5 ⩽ M , 2 N ⩽ M ) {\mathbb {C}^M}(2 \leqslant N,5 \leqslant M,2N \leqslant M) . There exists then a proper holomorphic immersion from D 1 {D_1} to D 2 {D_2} . Furthermore if P I ( D 1 , D 2 ) {\mathbf {PI}}({D_1},{D_2}) is the set of proper holomorphic immersions from D 1 {D_1} to D 2 {D_2} and A ( D 1 , D 2 ) A({D_1},{D_2}) is the set of holomorphic maps from D 1 {D_1} to D 2 {D_2} that are continuous on the boundary, then the closure of P I ( D 1 , D 2 ) {\mathbf {PI}}({D_1},{D_2}) in the topology of uniform convergence on compacta contains A ( D 1 , D 2 ) A({D_1},{D_2}) . The approximating proper maps can be made tangent to any finite order of contact at a given point. The same result was obtained for proper holomorphic maps, in one codimension, when the target domain has a plurisubharmonic 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) when the target domain is bounded weakly pseudoconvex with C 2 {C^2} -smooth boundary. If the target domain is strongly pseudoconvex then the approximating proper holomorphic maps can also be made continuous on the boundary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in C L {\mathbb {C}^L} is called " M M dimensional-pseudoconvex" (where L ⩾ M L \geqslant M ) if it has a smooth exhaustion function r r such that every point w w in this domain has some M M -dimensional complex affine subspace going through this point for which r r , restricted to this subspace, is strictly plurisubharmonic in w w . In the result mentioned above the assumption that the target domain is pseudoconvex in C M ( M ⩾ 2 N , 5 ) {\mathbb {C}^M}(M \geqslant 2N,5) can be substituted for the assumption that the domain is " M M -dimensional-pseudoconvex". Similarly, the assumption that the target domain D 2 {D_2} is " ( N + 1 ) (N + 1) -dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are " ( N + 1 ) (N + 1) -dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from D 1 {D_1} to D 2 {D_2} contains A ( D 1 , D 2 ) A({D_1},{D_2}) . All the results are obtained with embeddings when the Euclidean dimensions are such that dim C ( D 2 ) ⩾ 2 dim C ( D 1 ) + 1 {\dim _\mathbb {C}}({D_2}) \geqslant 2{\dim _\mathbb {C}}({D_1}) + 1 . Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from D 1 {D_1} to D 2 {D_2} contains A ( D 1 , D 2 ) A({D_1},{D_2}) .