  Say S is a compact three-manifold with non-positive Yamabe invariant. We
prove that in any long time constant mean curvature Einstein flow over S,
having bounded C^{\alpha} space-time curvature at the cosmological scale, the
reduced volume (-k/3)^{3}Vol(g(k)) (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, under the given assumptions, there is volume collapse in the
regions where the injectivity radius collapses (i.e. tends to zero) in the long
time. We conjecture that under the curvature assumption above the Thurston
geometrization is the unique global attractor. We validate it in some special
cases.
