The reduced Einstein equations and the conformal volume collapse of 3-manifolds (Q2769503)

This is a mathematical investigation of the dynamical system described by the reduced Einstein equations. The reduced Einstein \((3+1)\)-equations arise when the constraint equations have been factored out and gauge conditions have been imposed. In this paper the authors study ``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 \(-1\).'' A manifold \(M\) is Yamabe type \(-1\) if the only constant scalar curvature Riemannian metrics, that \(M\) admits, have negative constant scalar curvature.NEWLINENEWLINENEWLINEUsing a conformal reduction process the authors ``find that the reduced Einstein flow is described by a time-dependent non-local dimensionless reduced Hamiltonian \(H_{\text{reduced}}\) which is strictly monotonically decreasing along any non-constant integral curve of the reduced Einstein system''. They then ``discuss the relationships between \(H_{\text{reduced}}\), the \(\sigma\)-constant of \(M\), the Gromov norm \(\|M \|\) and the hyperbolic \(\sigma\)-conjecture''.NEWLINENEWLINENEWLINEAs an example the authors ``consider Bianchi models that spatially compactify to Yamabe type \(-1\)''. They find ``that under the reduced Einstein flow, \(H_{\text{reduced}}\), asymptotically approaches either the \(\sigma\)-constant or, in the hyperbolizable case, the conjectured \(\sigma\)-constant. \dots{} 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''. They also ``consider applications of these results to future all-time small data existence theorems for spatially compact spacetimes''.