  In this article we investigate the restrictions imposed by the dominant
energy condition (DEC) on the topology and conformal type of \textsl{possibly
non-compact} marginally outer-trapped surfaces (thus extending Hawking's
classical theorem on the topology of black holes). We first prove that an
unbounded, stable marginally outer trapped surface in an initial data set
$(M,g,k)$ obeying the dominant energy condition is conformally diffeomorphic to
either the plane $\mathbb{C}$ or to the cylinder $\mathbb{A}$ and in the latter
case infinitesimal rigidity holds. As a corollary, when the DEC holds strictly
this rules out the existence of trapped regions with cylindrical boundary. In
the second part of the article, we restrict our attention to asymptotically
flat data $(M,g,k)$ and show that, in that setting, the existence of an
unbounded, stable marginally outer trapped surface essentially never occurs
unless in a very specific case, since it would force an isometric embedding of
$(M,g,k)$ into the Minkowski spacetime as a space-like slice.
