  Let (M,g) be a complete 3-dimensional asymptotically flat manifold with
everywhere positive scalar curvature. We prove that, given a compact subset K
of M, all volume preserving stable constant mean curvature surfaces of
sufficiently large area will avoid K. This complements the work of G. Huisken
and S.-T. Yau and J. Qing and G. Tian whose results show that large spherical
volume preserving stable constant mean curvature spheres that omit a large
compact subset of an asymptotically flat manifold that is $C^4$-asymptotic to
positive mass Schwarzschild are uniquely determined by their area. The analysis
of Huisken and Yau as well as that of Qing and Tian takes place in the
asymptotic regime of M. Here we adapt ideas from the minimal hypersurface proof
of the positive mass theorem 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.
