Constant mean curvature foliations in cosmological spacetimes

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