Constant mean curvature foliations in cosmological spacetimes

摘要: Foliations by constant mean curvature hypersurfaces provide a possibility of defining preferred time coordinate in general relativity. In the following various conjectures are made about existence foliations this kind spacetimes satisfying strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these under supplementary assumptions is reviewed. The method proof used explained prospects for generalizing it discussed. relations questions to cosmic censorship closed universe recollapse conjecture pointed out. 1. Conjectures purpose paper put forward some spatially present recent results which show that true certain special cases. considered globally hyperbolic solutions Einstein equations with vanishing cosmological possess hypersurface satisfy condition, i.e. Tαβu u + 1 2 T α ≥ 0 any unit timelike vector u. Following terminology Bartnik [1], I refer as spacetimes. If S spacelike spacetime (M, gαβ), its induced metric second fundamental form will be denoted gab kab respectively. trace trk = gkab. said have (CMC) if function constant. There number well-known properties CMC (see e.g. [2]). first concerns uniqueness: there exists at most one given non-zero value trk. case maximal (trk 0) statement not quite so strong. unless static Killing t Rαβt 0. For coupled reasonable matter latter situation rare. instance, satisfies non-negative pressures (Tαβx x x) dominant then type necessarily flat. next property spacetime, foliation neighbourhood