## 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

### MSC:

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