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Remarks on cosmological spacetimes and constant mean curvature surfaces. (English) Zbl 0647.53044

A Lorentzian manifold (M,g) is called a cosmological spacetime if it is globally hyperbolic with compact Cauchy surfaces and satisfies the timelike convergence condition Ric(T,T)\(\geq 0\), for every timelike vector T. Here the existence of constant mean curvature Cauchy surfaces for cosmological spacetimes is investigated. It is shown that such a surface exists through \(p\in M\) if \(M\setminus I(p)\) is compact. Examples are given for cosmological spacetimes admitting no Cauchy surface of constant mean curvature, but having a non-Cauchy surface of that type.
Furthermore the investigations are used to prove a strengthening of G. J. Galloway’s splitting theorem [Commun. Math. Phys. 96, 423-429 (1984; Zbl 0575.53040)]. But as indicated by the author, this also can be derived from a very recent splitting theorem due to J.-H. Eschenburg [see the preceding review].
Reviewer: Bernd Wegner

MSC:

53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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