Vacuum Space-Times That Admit No Maximal Slice

Every closed three-manifold occurs as a spacelike hypersurface of a vacuum space-time. For most of these three-manifolds, however, the vacuum space-times that contain them have no maximal slice. For asymptotically flat manifolds there are no restrictions on which three-manifolds can occur obeying the local energy conditions $\rho >{\left(J_a{J}^a\right)}^{\frac{1}{2}}$ and the space-times that contain them in most cases have no maximal slice.