On the Hartogs extension theorem for unbounded domains in $\mathbb{C}^n$

TL;DR

This paper characterizes when unbounded domains in complex space allow holomorphic extension of CR functions, linking it to the envelope of holomorphy of the complement, thus providing a geometric criterion for the Hartogs phenomenon.

Contribution

It provides the first geometric characterization of unbounded domains in ^n for which the Hartogs extension property holds.

Findings

Extension property holds iff the envelope of holomorphy of the complement is ^n.

Extension failure can occur if the domain contains a complex hypersurface.

The result depends on the boundary's contact geometry with complex hypersurfaces.

Abstract

Let $\Omega\subset\mathbb{C}^n$, $n\geq 2$, be a domain with smooth connected boundary. If $\Omega$ is relatively compact, the Hartogs-Bochner theorem ensures that every CR distribution on $\partial\Omega$ has a holomorphic extension to $\Omega$. For unbounded domains this extension property may fail, for example if $\Omega$ 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 $\mathbb{C}^n\backslash\overline{\Omega}$ is $\mathbb{C}^n$. It seems that it is a first result in the literature which gives a geometric characterization of unbounded domains in $\mathbb C^n$ for which the Hartogs phenomenon holds. Comparing this to earlier work by the first two authors and Z.~S{\l}odkowski, one observes that the extension problem sensitively depends on a finer geometry of the contact of a complex hypersurface and the boundary of the domain.