Abstract
We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, by which we mean globally hyperbolic spacetimes with compact Cauchy surfaces. If a cosmological spacetime satisfying the strong energy condition contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant Λ > 0 \Lambda > 0 . Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.