Cosmological time versus CMC time II: the de Sitter and anti-de Sitter cases
Résumé
This paper continues the investigation of constant mean curvature (CMC) time functions in maximal globally hyperbolic spatially compact spacetimes of constant sectional curvature, which was started in math.DG/0604486. In that paper, the case of flat spacetimes was considered, and in the present paper, the remaining cases of negative curvature (i.e. anti-de Sitter) spacetimes and postitive curvature (i.e. de Sitter) spacetimes is dealt with. As in the flat case, the existence of CMC time functions is obtained by using the level sets of the cosmological time function as barriers. A major part of the work consists of proving the required curvature estimates for these level sets. The nonzero curvature case presents significant new difficulties, in part due to the fact that the topological structure of nonzero constant curvature spacetimes is much richer than that of the flat spacetimes. Further, the timelike convergence condition fails for de Sitter spacetimes, and hence uniqueness for CMC hypersurfaces fails in general. We characterize those de Sitter spacetimes which admit CMC time functions (which are automatically unique), as well as those which admit CMC foliations but no CMC time function.