  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.
