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

Abstract

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 H reduced which is strictly monotonically decreasing along any non-constant integral curve of the reduced Einstein system. We establish relationships between H reduced, the σ-constant of M, and the Gromov norm ‖M‖, show that H reduced has a global minimum at a hyperbolic critical point if and only if the hyperbolicσ-conjecture is true, and show that for rigid hyperbolizable M, the hyperbolic fixed point of the reduced Einstein flow is a local attractor. We consider as examples Bianchi models that spatially compactify to manifolds of Yamabe type −1 and show that for the non-hyperbolizable models, the reduced Einstein flow volume-collapses the 3-manifold M along either circular fibers, embedded tori, or completely to a point, as suggested by conjectures in 3-manifold topology. Remarkably, in each of these cases of collapse, the collapse occurs with bounded curvature.