The Reduced Einstein Equations - a Brief Summary

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. We discuss relationships between the reduced Hamiltonian, the σ-constant of M, the Gromov norm ǁMǁ, and the phase portrait of the reduced Einstein equations. We consider as examples Bianchi models that spatially compactify to manifolds of negative Yamabe type and find that in the non-hyperbolizable cases, the conformal volume of the reduced Einstein flow volume collapses M along either circular fibers, embedded tori, or totally collapses M, precisely as occurs in the theory of collapsing Riemannian manifolds.