On the Hartogs Extension Theorem for Unbounded Domains in Cⁿ

Author

Al Boggess
Roman Dwilewicz, Missouri University of Science and TechnologyFollow
Egmont Porten

Abstract

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.

Recommended Citation

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

Department(s)

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)

0373-0956

Document Type

Article - Journal

Document Version

Final Version

File Type

text

Language(s)

English

Rights

© 2023 The Authors, All rights reserved.

Creative Commons Licensing

This work is licensed under a Creative Commons Attribution-No Derivative Works 4.0 License.

Publication Date

01 Jan 2022