The authors report on recent results about global properties of the Einstein gravitational field equations with incoherent matter. They restrict to a special class of metrics where the spatial part is a compact 3-manifold of positive signature, and the time coordinate is globally defined.
The plane symmetric and the spherically symmetric case are already often discussed in the literature, therefore, the present authors concentrate on such spatial parts, which can be represented by a cross product between a circle and a compact 2-manifold whose universal covering space is the plane of negative constant curvature.
They carefully discuss the question under which circumstances, the space-time can be covered by hypersurfaces of constant mean curvature.