Vacuum Spacetimes with Future Trapped Surfaces

We show that one can construct initial data for the Einstein equations which satisfy the vacuum constraints. This initial data are defined on a manifold with topology with a regular centre and are asymptotically flat. Furthermore, these initial data will contain a shell-like region which is foliated by two-surfaces of topology . These two-surfaces are future trapped in the language of Penrose. The Penrose singularity theorem guarantees that the vacuum spacetime which evolves from this initial data is future null incomplete.