Title:

Spacelike hypersurfaces of constant mean curvature

The objects under discussion in this thesis are spacelike hypersurfaces on a four dimensional pseudoRiemannian manifold. Various differentiability classes of such hypersurfaces are discussed and are made into topological spaces. The hypersurfaces* are taken to be topologically either B_{3} or S_{3}, in the latter case the manifold is taken to be topologically S_{3} x I, I ≤ ℝ an interval. In chapter I the preliminary definitions are given together with some topological results on the existence and location of such hypersurfaces. These properties and the compactness of certain spaces of hypersurfaces are related to the underlying topological and causality properties of the pseudoRiemannian manifold. In order to do this sufficient conditions are found for the compactness of the domain of dependence of a compact spacelike hypersurface. The main results of this chapter are theorems, (1.16), (1.28), (1.32), which state that a spacelike hypersurface with boundary in an asymptotically simple and empty, stably causal, manifold has a compact domain of dependence and that there is a neighbourhood of Minkowski space in the Whitney C^{3} fine topology of manifolds with these properties, further the closure of the space of C_{0} spacelike hypersurfaces with a boundary B in the space of C_{0} hypersurfaces with boundary B on such a manifold is compact with respect to the supremum topology. In chapter II various functionals on the spaces of spacelike hypersurfaces are discussed with respect to their continuity properties and a relationship is found between the geometrical mean curvature of a hypersurface and its property of extremising a linear combination of the area and hypovolume functionals. The area and hypovolume functionals are defined first for C_{1} and C_{0} hypersurfaces respectively and then their definitions are extended to the compact space of hypersurfaces discussed in chapter I  compact unless otherwise stated. The principal results of this chapter are theorems (2.10), (2.15.3), (2.24), in which it is shown that a C_{2} extremal of the functional A + _{K}V has constant mean curvature K and that both the area functional, A, and its extension, Ã, are upper semicontinuous with respect to the supremum topology. The variational result is then used to define the notion of generalised constant mean curvature and the compact spaces of spacelike hypersurfaces, discussed in chapter I, are shown to contain members of generalised constant mean curvature, by theorems (2.29), (2.30). The concept of a foliation of a region, or a manifold, by spacelike hypersurfaces is introduced in chapter III and the properties of such foliations whose leaves are hypersurfaces of constant mean curvature are discussed. It is shown that the constant mean curvature π can be used as a natural time coordinate unless the manifold is static, for stably causal manifolds which are topologically S^{3} x I , provided that Hawking's timelike convergence condition holds. It is further shown that a C_{1} spacelike hypersurface of constant mean curvature π is always a local maximum of the functional A+πV and is a strict local maximum if the hypersurface is part of a foliation on which π is a strictly monotonic increasing function of time with > O. A relationship is found between the mean curvatures of two touching, spacelike hypersurfaces at their point of contact. This result is then used to show that any C_{2} foliation of a manifold by spacelike hypersurfaces without boundary of constant mean curvature is unique and that there are no such hypersurfaces which are not members of the given foliation. The principal results of this chapter are theorems, (3.9), (3.23.1), (3.24), (3.26). In chapter IV the properties of the one parameter families of spacelike hypersurfaces of generalised constant mean curvature which were shown to exist at the end of chapter II are discussed. It is found, with the additional assumptions that there is one and only one such spacelike hypersurface for each value of the generalised constant mean curvature K , and that for K_{1} K_{2}, S_{K1} S_{K2} (where S_{Ki} is the spacelike hypersurface of generalised constant mean curvature K_{i}), that these families can be considered as the leaves of foliations with many of the properties ofthe foliations discussed in chapter III. In particular, if the hypersurfaces S_{Ki} maximise the functionals A+_{Ki}V and if K_{1} > K_{2} then S_{K2} ≤ J^{}T~[S_{K1}] : essentially this is the result that K> increases monotonically with time. It is also shown that the region between two such hypersurfaces is foliated by the hypersurfaces S_{K} with K ϵ [K_{1},K_{2}], in the sense that at least one such hypersurface passes through each point in this region and that every stably causal, topologically S^{3} x ℝ manifold is foliated in this sense by its family of S_{K}, if it is presumed to have an S_{K} for each K ϵ ℝ. The principal results of this chapter are theorems, (4.6), (4.8), (4.10)). Chapter V consists of a discussion of noncompact spacelike hypersurfaces of constant mean curvature. Bernstein's theorem and some related concepts are investigated; there follows a short discussion of some limited existence theorems for such hyper surf aces. There is an appendix about some of the applications to physics of the results above. The possibility of using mean curvature as a naturally defined cosmic time is discussed as is the problem of positivity of gravitational energy.
