Spacelike hypersurfaces of constant mean curvature and Calabi-Bernstein type problems

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson–Walker spacetimes

“…In particular, they found that, when the ambient space obeys the so called timelike convergence condition, then every compact spacelike hypersurface with constant mean curvature must be totally umbilical and, except in very exceptional cases, it must be a spacelike slice. As observed by those same authors in [5], the weaker null convergence condition is enough to guarantee that the hypersurface must be totally umbilical, not only for hypersurfaces into GRW spacetimes but, more generally, for hypersurfaces into CS spacetimes equipped with a timelike conformal vector field which is an eigenfield of the Ricci operator (and, in particular, for CS spacetimes which are equipped with a closed timelike conformal vector field). Later on, Montiel [28] considered again that question and, after a careful classification of the totally umbilical hypersurfaces with constant mean curvature, he completed the program by showing that the only compact spacelike hypersurfaces with constant mean curvature into a GRW spacetime which satisfies the null convergence condition are the spacelike slices, unless in the case where the spacetime is a de Sitter space and the hypersurface is a round umbilical sphere.…”

Integral Formulae for Spacelike Hypersurfaces in Conformally Stationary Spacetimes and Applications

“…Specifically, in [7] they obtained the first and second Minkowski formulae for compact spacelike hypersurfaces in conformally stationary spacetimes, and applied them to the study of compact spacelike hypersurfaces with constant mean curvature (see also [5,6] for a first version of those formulae in the case where the ambient spacetime is a generalized Robertson-Walker spacetime). More recently, Montiel [26] has given another proof of the first and second Minkowski formulae in the case where the ambient spacetimeM is equipped with a conformal timelike vector field K which is also closed, in the sense that its metrically equivalent 1-form is closed.…”

“…The use of those kinds of integral formulae in the Lorentzian setting was started by Montiel in [25] for the case of spacelike hypersurfaces with constant mean curvature in de Sitter space, and it was continued by the first author together with Romero and Sánchez for the case of spacelike hypersurfaces with constant mean curvature in more general spacetimes (generalized Robertson-Walker spacetimes [5,6] and, more generally, conformally stationary spacetimes [7]). Observe that for the case of the mean curvature only the first and the second Minkowski formulae are needed.…”

“…As for the case of more general Lorentzian ambient spaces, in a series of recent papers, the first author together with Romero and Sánchez [5][6][7] studied the uniqueness of space-like hypersurfaces with constant mean curvature in a wide class of Lorentzian manifolds, the so-called conformally stationary spacetimes. Since our work will be developed in such ambient spacetimes, we will describe here that class of Lorentzian manifolds, and its relation to other known spacetimes, such as generalized Robertson-Walker spacetimes.…”

Special standard static space–times

“…The problem of existence of a warping function which makes the warped product Einstein was already studied for special cases such as generalized Robertson-Walker space-times and a table given summarizing different cases of Einstein Ricci tensor of a generalized Robertson-Walker when the Ricci tensor of the fiber is Einstein in [1] (see also references therein). In this paper, we consider this problem for standard static space-times.…”