Abstract
The existence of hypersurfaces of constant mean extrinsic curvature is examined. Using techniques developed by Choquet-Bruhat in her work on related subjects and techniques used by D'Eath in his study of perturbed Robertson-Walker universes, theorems are proved about the existence of slices of constant mean extrinsic curvature for spacetimes in a neighbourhood of the open Robertson-Walker Universes. It is shown in particular that those spacetimes which lie in a neighbourhood of Minkowski space or de-Sitter space admit slices of constant mean extrinsic curvature. By modifying the techniques used to prove these theorems, it is shown that asymptotically simple spacetimes which are close to Minkowski space admit slices of constant mean extrinsic curvature. The behaviour of these slices near null infinity is examined and it is shown that a large family of such hypersurfaces exists, indexed by the BMS supertranslations.
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