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Transactions of the American Mathematical Society

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Immersions and embeddings in domains of holomorphy

Author: Avner Dor
Journal: Trans. Amer. Math. Soc. 347 (1995), 2813-2849
MSC: Primary 32H02; Secondary 32D05
MathSciNet review: 1282885
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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})$.

References [Enhancements On Off] (What's this?)

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