Cosmological time versus CMC time in spacetimes of constant curvature (Q431568)

Let \(M\) be a maximal globally hyperbolic space-time of constant curvature \(k\) with compact space-like Cauchy hypersurface. The authors study the existence of foliations of \(M\) by constant mean curvature (CMC) hypersurfaces. In the case of non-positive curvature \(k\), they prove the existence of a global foliation of \(M\) by CMC Cauchy hypersurfaces. In the case of positive curvature \(k\), they get only a local result : the existence of a foliation by Cauchy hupersurfaces in a neighborhood of the past (or future) singularity. Except from some special cases, a CMC foliation is constructed as a level set of a CMC time function such that the (constant) mean curvature of a level set increases with time. The construction of a CMC time function is based on a notion of asymptotic barrier. For \(c \in \mathbb{R}\), a pair of \(c\)-barriers is a pair of space-like \(C^0\)-Cauchy hypersurfaces \(\Sigma^{\pm}\) such that \(\Sigma^+\) is in the future of \(\Sigma^-\) and the corresponding mean curvatures \(H_{\Sigma^{\pm}}\) are related by \(H_{\Sigma^-} \leq c \leq H_{\Sigma^{+}}\). An appropriate sequence of asymptotic barriers allows to construct a CMC time function.