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arXiv · 0806.0669

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

Martin Reiris
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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.

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Martin Reiris. 2008-06-04. On the asymptotic spectrum of the reduced volume in cosmological solutions of the Einstein equations. https://doi.org/10.1007/s10714-008-0693-6

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