Foliations of space-times by spacelike hypersurfaces of constant mean curvature
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The foliations under discussion are of two different types, although in each case the leaves areC2 spacelike hypersurfaces of constant mean curvature. For manifolds, such as that of the Friedmann universe with closed spatial sections, which are topologicallyI×S3,I an open interval, the leaves will be spacelike hypersurfaces without boundary and the foliation will fill the manifold. In the case of the domain of dependence of a spacelike hypersurface,S, with boundaryB, the leaves will be spacelike hypersurfaces with boundary,B, and the foliation will fillD(S).
It is shown that a local energy condition ensures that the constant mean curvature increases monotonically with time through such foliations and that, in the case of a foliation whose leaves are spacelike hypersurfaces without boundary in a manifold where this energy condition is satisfied globally, the foliation is unique.
KeywordsNeural Network Manifold Statistical Physic Complex System Energy Condition
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