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Asymptotic behaviour of future-complete cosmological spacetimes. (English) Zbl 1040.83041

The author compares two approaches about the global structure of a cosmological spacetime \((M,g)\) endowed with a compact mean curvature (CMC) Cauchy surface \((\Sigma, g,K)\). Let \((M,g)\) be a globally hyperbolic spacetime, with a foliation \(\mathcal{F}=\{\Sigma_\tau\}\) by CMC Cauchy surfaces, parametrized by their mean curvature \(\tau\). It is assumed that \((M,g)\) is a solution of the vacuum Einstein equations \(Ric =0\) and that the foliation \(\mathcal{F}\) is global in that \(M=M_\mathcal{F}\) and that \((M,g)\) is geodesically complete to the future of an initial slice \(\Sigma=\Sigma_{\tau_0}\). The author obtains some monotonicity formulae, some results about mean curvature rescaling and proper time rescaling and the structure and existence of limits. Next he studies the spaces of metrics with bounded curvature, the asymptotics and geometrization of \(3\)-manifolds, and the relations with the Sigma constant.

MSC:

83F05 Relativistic cosmology
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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