Immersions and embeddings in domains of holomorphy

Abstract:

Let ${D_1}$ be a bounded smooth strongly pseudoconvex domain in ${\mathbb {C}^N}$ and let ${D_2}$ be a domain of holomorphy in ${\mathbb {C}^M}(2 \leqslant N,5 \leqslant M,2N \leqslant M)$. There exists then a proper holomorphic immersion from ${D_1}$ to ${D_2}$. Furthermore if ${\mathbf {PI}}({D_1},{D_2})$ 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 continuous on the boundary, then the closure of ${\mathbf {PI}}({D_1},{D_2})$ in the topology of uniform convergence on compacta contains $A({D_1},{D_2})$. 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 ${C^2}$-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 ${\mathbb {C}^L}$ is called "$M$dimensional-pseudoconvex" (where $L \geqslant M$) if it has a smooth exhaustion function $r$ such that every point $w$ 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 $w$. In the result mentioned above the assumption that the target domain is pseudoconvex in ${\mathbb {C}^M}(M \geqslant 2N,5)$ can be substituted for the assumption that the domain is "$M$-dimensional-pseudoconvex". Similarly, the assumption that the target domain ${D_2}$ is "$(N + 1)$-dimensional-pseudoconvex" and all 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 ${D_2}$ contains $A({D_1},{D_2})$. All the results are obtained with embeddings when the Euclidean dimensions are such that ${\dim _\mathbb {C}}({D_2}) \geqslant 2{\dim _\mathbb {C}}({D_1}) + 1$. Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from ${D_1}$ to ${D_2}$ contains $A({D_1},{D_2})$.