Surfaces with maximal constant mean curvature

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.