On large volume preserving stable constant mean curvature surfaces in asymptotically flat manifolds

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.