  This paper gives a new proof that maximal, globally hyperbolic, flat
spacetimes of dimension $n\geq 3$ with compact Cauchy hypersurfaces are
globally foliated by Cauchy hypersurfaces of constant mean curvature, and that
such spacetimes admit a globally defined constant mean curvature time function
precisely when they are causally incomplete. The proof, which is based on using
the level sets of the cosmological time function as barriers, is conceptually
simple and will provide the basis for future work on constant mean curvature
time functions in general constant curvature spacetimes, as well for an
analysis of the asymptotics of constant mean foliations.
