Constant mean curvature foliations of flat space-times

Constant Mean Curvature Foliations of Simplicial Flat Spacetimes

Benedetti and Guadagnini [5] have conjectured that the constant mean curvature foliation Mτ in a 2 + 1 dimensional flat spacetime V with compact hyperbolic Cauchy surfaces satisfies limτ→−∞ lMτ = sT , where lMτ and sT denote the marked length spectrum ofMτ and the marked measure spectrum of the R-tree T , dual to the measured foliation corresponding to the translational part of the holonomy of ...

Foliations by constant mean curvature tubes

Let Γ be a nondegenerate geodesic in a compact Riemannian manifold M . We prove the existence of a partial foliation of a neighbourhood of Γ by CMC surfaces which are small perturbations of the geodesic tubes about Γ. There are gaps in this foliation, which correspond to a bifurcation phenomenon. Conversely, we also prove, under certain restrictions, that the existence of a partial CMC foliatio...

Constant mean curvature foliations in cosmological spacetimes

Foliations by constant mean curvature hypersurfaces provide a possibility of defining a preferred time coordinate in general relativity. In the following various conjectures are made about the existence of foliations of this kind in spacetimes satisfying the strong energy condition and possessing compact Cauchy hypersurfaces. Recent progress on proving these conjectures under supplementary assu...

Hypersurfaces with Constant Mean Curvature in a Lorentzian Space Form

Bel–robinson Energy and Constant Mean Curvature Foliations

An energy estimate is proved for the Bel–Robinson energy along a constant mean curvature foliation in a spatially compact vacuum spacetime, assuming an L∞ bound on the second fundamental form, and a bound on a spacetime version of Bel–Robinson energy.