  In this note we consider asymptotically flat manifolds with non-negative
scalar curvature and an inner boundary which is an outermost minimal surface.
We show that there exists an upper bound on the mean curvature of a constant
mean curvature surface homologous to a subset of the interior boundary
components. This bound allows us to find a maximizer for the constant mean
curvature of a surface homologous to the inner boundary.
  With this maximizer at hand, we can construct an increasing family of sets
with boundaries of increasing constant mean curvature. We interpret this
familiy as a weak version of a CMC foliation.
