Another attractive time coordinate is constant mean curvature (CMC) time. For a general discussion of this see . A global existence theorem in this time for spacetimes with two Killing vectors and certain matter models (collisionless matter, wave maps) was proved in . That the choice of matter model is important for this result was demonstrated by a global non-existence result for dust in . As shown in , this leads to examples of spacetimes that are not covered by a CMC slicing. Results on global existence of CMC foliations have also been obtained for spherical and hyperbolic symmetry [206, 51].
A drawback of the results on the existence of CMC foliations just cited is that they require as a hypothesis the existence of one CMC Cauchy surface in the given spacetime. More recently, this restriction has been removed in certain cases by Henkel using a generalization of CMC foliations called prescribed mean curvature (PMC) foliations. A PMC foliation can be built that includes any given Cauchy surface  and global existence of PMC foliations can be proved in a way analogous to that previously done for CMC foliations [131, 132]. These global foliations provide barriers that imply the existence of a CMC hypersurface. Thus, in the end it turns out that the unwanted condition in the previous theorems on CMC foliations is in fact automatically satisfied. Connections between areal, CMC, and PMC time coordinates were further explored in . One important observation there is that hypersurfaces of constant areal time in spacetimes with symmetry often have mean curvature of a definite sign.
Once global existence has been proved for a preferred time coordinate, the next step is to investigate the asymptotic behaviour of the solution as . There are few cases in which this has been done successfully. Notable examples are Gowdy spacetimes [84, 139, 87] and solutions of the Einstein-Vlasov system with spherical and plane symmetry . Progress in constructing spacetimes with prescribed singularities will be described in Section 6 . In the future this could lead in some cases to the determination of the asymptotic behaviour of large classes of spacetimes as the singularity is approached.
|Theorems on Existence and Global Dynamics for the
Alan D. Rendall
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