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

TL;DR

This paper studies the long-term behavior of cosmological solutions to Einstein's equations on certain three-manifolds, showing that the reduced volume decreases monotonically and suggesting a link to Thurston geometrization.

Contribution

It proves the monotonic decay of the reduced volume in Einstein flows with bounded curvature on manifolds with non-positive Yamabe invariant, connecting geometric flow behavior to Thurston geometrization.

Findings

Reduced volume decays monotonically over time.

Volume collapse occurs where injectivity radius tends to zero.

Thurston geometrization conjecture is a potential global attractor under certain conditions.

Abstract

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.