On the boundary of the region containing trapped surfaces

The boundary of the region in spacetime containing future-trapped closed surfaces is considered. In asymptotically flat spacetimes, this boundary does not need to be the event horizon nor a dynamical/trapping horizon. Some properties of this boundary and its localization are analyzed, and illustrated with examples. In particular, fully explicit future-trapped compact surfaces penetrating into flat portions of a Vaidya spacetime are presented.