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Maximal hypersurfaces and foliations of constant mean curvature in general relativity

Marsden, Jerrold E. and Tipler, Frank J. (1980) Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Physics Reports, 66 (3). pp. 109-139. ISSN 0370-1573. doi:10.1016/0370-1573(80)90154-4.

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We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to space times that are vacuum and non-flat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe: i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez' original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions: we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat. Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.

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Additional Information:© 1980 North-Holland Publishing Company. Received May 1980. Available online 23 September 2002.
Issue or Number:3
Record Number:CaltechAUTHORS:20100910-092318457
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19861
Deposited By: Ruth Sustaita
Deposited On:10 Sep 2010 16:42
Last Modified:08 Nov 2021 23:55

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