Embedding strictly pseudoconvex domains into balls. (English) Zbl 0594.32024

This paper contains a number of interesting results on proper holomorphic mappings from a strictly pseudoconvex domain D to a (higher-dimensional) ball \({\mathbb{B}}^ N\). The first result is that there are domains D with smooth real-analytic boundary such that no proper mapping \(f: D\to {\mathbb{B}}^ n\) extends smoothly to \(\bar D.\) (A similar result has also been obtained by J. Faran.) The next result is a strengthening of an embedding theorem of Fornaess and Henkin: D can be nicely embedded into a bounded strictly convex domain with real analytic boundary. The third result is that for any holomorphic mapping \(h=(h_ 1,...,h_ p): D\to {\mathbb{C}}^ p\) with \(h\in C(\bar D)\) and \(h(\bar D)\subset {\mathbb{B}}^ p\), there are holomorphic functions \(f_ 1,...,f_ s\) such that \((h_ 1,...,h_ p,f_ 1,...,f_ s)\) maps D properly to \({\mathbb{B}}^{p+s}\). (A similar result has also been obtained by E. Low.)
Reviewer: E.Bedford


32V40 Real submanifolds in complex manifolds
32H35 Proper holomorphic mappings, finiteness theorems
32T99 Pseudoconvex domains
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