EXISTENCE OF CONSTANT MEAN-CURVATURE FOLIATIONS IN SPACETIMES WITH 2-DIMENSIONAL LOCAL SYMMETRY

It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvatu re hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves o f this foliation takes on arbitrarily negative values and so the initi al singularity in these spacetimes is a crushing singularity. The simp lest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are a lso covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry sin gles out a compact surface passing through any given point of spacetim e and the Hawking mass of any such surface is non-negative. If the Haw king mass of any one of these surfaces is zero then the entire spaceti me is flat.