The Einstein flow, the sigma-constant and the geometrization of 3-manifolds

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Abstract
For the problem of the Hamiltonian reduction of Einstein's equations on a (3 + 1) vacuum spacetime that admits a foliation by constant mean curvature (CMC) compact spacelike hypersurfaces M that satisfy certain topological restrictions, we introduce a dimensionless non-local time-dependent reduced Hamiltonian system where and Preduced is the reduced phase space. For this system, we establish the following properties: (1) Hreduced has a unique critical point at the hyperbolic point which is a strict local minimum; (2) Hreduced is a strictly monotonically decreasing function along the Einstein flow unless , in which case Hreduced is constant in time; and (3) the infimum of Hreduced is related to the -constant of M by Further applications and developments relating the Einstein flow to (M) and to the geometrization conjecture for 3-manifolds are discussed.

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