Let (M, g) be a complete 3-dimensional asymptotically flat manifold with everywhere positive scalar curvature. We prove that,
given a compact subset K M, all volume preserving stable constant mean curvature surfaces of sufficiently large area will avoid
K. This complements the results of G. Huisken and S.-T. Yau [17] and of J. Qing and G. Tian [26] on the uniqueness of large volume
preserving stable constant mean curvature spheres in initial data sets that are asymptotically close to Schwarzschild with mass
m > 0. The analysis in [17] and [26] takes place in the asymptotic regime of M. Here we adapt ideas from the minimal surface proof
of the positive mass theorem [32] by R. Schoen and S.-T. Yau and develop geometric properties of volume preserving stable constant
mean curvature surfaces to handle surfaces that run through the part of M that is far from Euclidean.