The reduced Hamiltonian of general relativity and the σ-constant of conformal geometry

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 H-reduced : R- X P-reduced -> R where the reduced Hamiltonian is given by H-reduced(tau,gamma ,p(TT)) = -tau (3) integral (M) phi (6) mu (gamma) = -tau (3) integral (M) mug = -tau (3)vol(M,g). For compact connected oriented 3-manifolds of Yamabe type -1, we establish the following properties for this reduced system: 1. H-reduced(tau,gamma ,p(TT)) is a monotonically decreasing function of t unless p(TT) = 0 and gamma = <(<gamma>)over bar> is hyperbolic, at which point H-reduced(tau, <(<gamma>)over tilde>, 0) is constant in time. 2. For tau epsilon R- fixed, H-reduced(tau,gamma ,p(TT)) has a unique (up to isometry) critical point at (<(<gamma>)over tilde>, 0) which is a strict local minimum tin the non-isometric directions). 3. For tau epsilon R- fixed, the sigma -constant of M is related to H-reduced by sigma (M) = -1/2 (((gamma ,pTT)epsilon Preduced) inf H-reduced (tau,gamma ,p(TT)))(2/3). If M is a hyperbolic manifold, then we conjecture that (<(<gamma>)over tilde>, 0) is a strict global minimum of H-reduced(tau,gamma ,p(TT)) which, as part of our work, is equivalent to the conjecture that the sigma -constant of M is realized by the unique hyperbolic geometry on M. If M is not a hyperbolic manifold, then the sigma -constant is never realized by a metric on M but is only approached as a limit. In this case, the Einstein flow seeks to attain the sigma -constant asymptotically insofar as the reduced Hamiltonian is monotonically seeking to decay to its infimum, although possible obstructions, such as the formation of black holes, may prevent any particular solution from approaching the sigma -constant asymptotically. Further applications and developments in higher dimensions are discussed.