Existence and non-existence results for global constant mean curvature foliations

Abstract

If (M,g) is a Lorentz manifold and S a spacelike hypersurface, let h and k denote the induced metric and second fundamental form of 5’ respectively. The mean curvature of S is the trace trhJi. An interesting class of spacelike hypersurfaces are those whose mean curvature is constant (CMC hypersurfaces). The Lorentz manifolds of primary interest in the following are those which possess a compact Cauchy hypersurface and whose Ricci tensor satisfies r(V, V) >_ 0 for any timelike vector V. They will be referred to as cosmological spacetimes. The curvature condition is known as the strong energy condition and implies certain uniqueness statements for CMC hypersurfaces. Under very general conditions, if a region of a cosmological spacetime is foliated by compact CMC hypersurfaces, then each of these has a different value of the mean curvature and a time coordinate t can be defined by the condition that its value at a point be equal to the mean curvature of the leaf of the foliation passing through that point. A time coordinate of this type will be referred to in the following as a CMC time coordinate. The questions, whether a cosmological spacetime contains a compact CMC hypersurface and which values the mean curvature can take on a hypersurface of this kind may seem purely geometrical in nature. However it turns out that the answers to these questions depend crucially on factors which have no obvious geometrical interpretation, but which have a physical meaning, when the Lorentz manifold is considered as a model for spacetime. The Einstein equations for a Lorentz metric g take the form G = 8xT, where G is the Einstein tensor of the metric g and T is the energy-momentum tensor. To get a determined system of evolution equations for the geometry and the matter, it is necessary to say more about the nature of the matter model used. This means specifying some matter fields, denoted collectively by b, a definition of T in terms of g and 4 and the differential equations which describe the dynamics of the matter. Putting these things together with the Einstein equations gives a system of evolution equations, the Einsteinmatter equations. It will be seen that the existence of global foliations by CMC hypersurfaces in a cosmological spacetime which is a solution of the Einsi.ein-matter equations depends essentially on the matter model chosen. Let s denote the scalar curvature of the metric g. The spacetime is said to satisfy the weak energy condition if r(V, V) > (1/2)sg(V,V) f or all timelike vectors V. (Note that, despite the terminology, the strong energy condition does not imply the weak one.) There is a topological obstruction to the existence of a compact spacelike hypersurface with vanishing mean curvature (maximal hypersurface) in a spacetime which satisfies the weak energy condition. In a cosmological spacetime this implies that if the Cauchy hypersurface is a manifold which admits no Riemannian metric with non-negative scalar curvature then the spacetime contains no compact maximal hypersurfaces. Moreover, there are strong restrictions in the case of a manifold which admits no Riemannian metric of positive scalar