It is shown that in a class of maximal globally hyperbolic spacetimes admitting two local Killing vectors, the past (defined with respect to an appropriate time orientation) of any compact constant mean curvature hypersurface can be covered by a foliation of compact constant mean curvature hypersurfaces. Moreover, the mean curvature of the leaves of this foliation takes on arbitrarily negative values and so the initial singularity in these spacetimes is a crushing singularity. The simplest examples occur when the spatial topology is that of a torus, with the standard global Killing vectors, but more exotic topologies are also covered. In the course of the proof it is shown that in this class of spacetimes a kind of positive mass theorem holds. The symmetry singles out a compact surface passing through any given point of spacetime and the Hawking mass of any such surface is non-negative. If the Hawking mass of any one of these surfaces is zero then the entire spacetime is flat.
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Similar content being viewed by others Explore related subjectsDiscover the latest articles and news from researchers in related subjects, suggested using machine learning. Author information Authors and AffiliationsInstitut des Hautes Etudes Scientifiques, 35 Route de Chartres, 91440 Bures sur Yvette, France, , , , , , FR
Alan D. Rendall
Received: 15 July 1996 / Accepted: 12 March 1997
About this article Cite this articleRendall, A. Existence of Constant Mean Curvature Foliations in Spacetimes with Two-Dimensional Local Symmetry . Comm Math Phys 189, 145–164 (1997). https://doi.org/10.1007/s002200050194
Issue Date: October 1997
DOI: https://doi.org/10.1007/s002200050194
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