Immersions and embeddings in domains of holomorphy

Author:
Avner Dor

Journal:
Trans. Amer. Math. Soc. **347** (1995), 2813-2849

MSC:
Primary 32H02; Secondary 32D05

DOI:
https://doi.org/10.1090/S0002-9947-1995-1282885-5

MathSciNet review:
1282885

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Abstract: Let be a bounded smooth strongly pseudoconvex domain in and let be a domain of holomorphy in . There exists then a proper holomorphic immersion from to . Furthermore if is the set of proper holomorphic immersions from to and is the set of holomorphic maps from to that are continuous on the boundary, then the closure of in the topology of uniform convergence on compacta contains . 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 -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 is called "dimensional-pseudoconvex" (where ) if it has a smooth exhaustion function such that every point in this domain has some -dimensional complex affine subspace going through this point for which , restricted to this subspace, is strictly plurisubharmonic in . In the result mentioned above the assumption that the target domain is pseudoconvex in can be substituted for the assumption that the domain is "-dimensional-pseudoconvex". Similarly, the assumption that the target domain is "-dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are "-dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from to contains .

All the results are obtained with embeddings when the Euclidean dimensions are such that . Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from to contains .

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DOI:
https://doi.org/10.1090/S0002-9947-1995-1282885-5

Article copyright:
© Copyright 1995
American Mathematical Society