  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.
