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On the Hartogs extension theorem for unbounded domains in Cn

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Author list: Porten, Egmont
Publication year: 2018
ISBN: 978-91-88527-87-5

Abstract

Let ? ? Cn, n = 2, be a domain with smooth connected boundary. If? is relatively compact, the Hartogs-Bochner theorem ensures that everyCR distribution on ?? has a holomorphic extension to ?. For unboundeddomains this extension property may fail, for example if ? contains a complex hypersurface. The main result in this paper tells that the extensionproperty holds if and only if the envelope of holomorphy of Cn\? is Cn.It seems that it is a first result in the literature which gives a geometriccharacterization of unbounded domains in Cnfor which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authorsand Z. S lodkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface andthe boundary of the domain.


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