Let Ω ⊂ Cn, n > 2, be a domain with smooth connected boundary. If Ω is relatively compact, the Hartogs–Bochner theorem ensures that every CR distribution on ∂Ω has a holomorphic extension to Ω. For unbounded domains this extension property may fail, for example if Ω contains a complex hypersurface. The main result in this paper tells that the extension property holds if and only if the envelope of holomorphy of Cn \ Ω is Cn. It seems that it is the first result in the literature which gives a geometric characterization of unbounded domains in Cn for which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authors and Z. Słodkowski, one observes that the extension problem changes in character if one restricts to CR functions of higher regularity.
A. Boggess et al., "On the Hartogs Extension Theorem for Unbounded Domains in Cn," Annales de l'Institut Fourier, vol. 72, no. 3, pp. 1185 - 1206, Association des Annales de l'Institut Fourier, Jan 2022.
The definitive version is available at https://doi.org/10.5802/aif.3514
Mathematics and Statistics
Keywords and Phrases
CR functions; envelopes of holomorphy; Hartogs–Bochner extension theorem; unbounded domains in Stein manifolds
International Standard Serial Number (ISSN)
Article - Journal
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01 Jan 2022