CMC foliation conjecture for globally hyperbolic spacetimes

A spacetime is maximally extended if it admits no further extension, and it is globally hyperbolic if it has a Cauchy surface. A constant mean curvature (CMC) hypersurface is a spacelike hypersurface whose mean curvature is constant. CMC foliation conjecture. Every maximally extended, globally hyperbolic spacetime can be foliated by CMC hypersurfaces. CMC foliations are important because they provide a time coordinate independent of spacetime symmetry and are closely related to the behavior of crushing singularities. The source gives no resolution of this conjecture.