On the asymptotic spectrum of the reduced volume in cosmological solutions of the Einstein equations

Let Σ be a compact three-manifold with non-positive Yamabe invariant. We prove that in any long time constant mean curvature Einstein flow over Σ, having bounded C ^α space–time curvature at the cosmological scale, the reduced volume [g(k) is the evolving spatial three-metric and k the mean curvature] decays monotonically towards the volume value of the geometrization in which the cosmologically normalized flow decays. In more basic terms, we show that there is volume collapse in the regions where the injectivity radius collapses (i.e. tends to zero) in the long time limit. We conjecture that under the curvature assumption above the Thurston geometrization is the unique global attractor. We prove this in some special cases.