Conformal Volume Collapse of 3-Manifolds and the Reduced Einstein Flow

摘要: We consider the problem of Hamiltonian reduction Einstein’s equations on a (3+1)-vacuum spacetime that admits foliation by constant mean curvature compact spacelike hypersurfaces M Yamabe type − 1. After conformal process, we find reduced Einstein flow is described time-dependent non-local dimensionless H which strictly monotonically decreasing along any non-constant integral curve system. establish relationships between reduced, σ-constant M, and Gromov norm ‖M‖, show has global minimum at hyperbolic critical point if only hyperbolicσ-conjecture true, for rigid hyperbolizable fixed local attractor. as examples Bianchi models spatially compactify to manifolds −1 non-hyperbolizable models, volume-collapses 3-manifold either circular fibers, embedded tori, or completely point, suggested conjectures in topology. Remarkably, each these cases collapse, collapse occurs with bounded curvature.