The reduced Einstein equations and the conformal volume collapse of 3-manifolds

Details The reduced Einstein equations and the conformal volume collapse of 3-manifolds The reduced Einstein equations and the conformal volume collapse of 3-manifolds

Nov 2001

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2001cqgra..18.4493f&link_type=abstract

Classical and Quantum Gravity, Volume 18, Issue 21, pp. 4493-4515 (2001).

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We consider the problem of the Hamiltonian reduction of Einstein's equations on a (3 + 1)-vacuum spacetime that admits a foliation by constant mean curvature compact spacelike hypersurfaces M of Yamabe type -1. After a conformal reduction process, we find that the reduced Einstein flow is described by a time-dependent non-local dimensionless reduced Hamiltonian Hreduced which is strictly monotonically decreasing along any non-constant integral curve of the reduced Einstein system. We discuss relationships between Hreduced, the σ-constant of M, the Gromov norm |M|| and the hyperbolic σ-conjecture. As examples, we consider Bianchi models that spatially compactify to manifolds of Yamabe type -1. For these models we show that under the reduced Einstein flow, Hreduced asymptotically approaches either the σ-constant or in the hyperbolizable case, the conjectured σ-constant, as suggested by our general theory. In the non-hyperbolizable cases, the conformal metric of the reduced Einstein flow volume-collapses M along either circular fibres, embedded tori, or collapses the entire manifold to a point, and in each case, the collapse occurs with bounded curvature. We consider applications of these results to future all-time small-data existence theorems for spatially compact spacetimes.

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