Abstract
We establish a new CMC (constant mean curvature) existence result for
cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact
Cauchy surfaces satisfying the strong energy condition. If the spacetime
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 $\Lambda > 0$. Lastly, we include some
comments concerning the future causal boundary for cosmological spacetimes
which pertain to the CMC conjecture of the authors.