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We study the isoperimetric structure of asymptotically flat Riemannian3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0.Refining an argument due to H. Bray we obtain an effective volume comparisontheorem in Schwarzschild. We use it to show that isoperimetric regions exist in(M, g) for all sufficiently large volumes, and that they are close to centeredcoordinate spheres. This implies that the volume-preserving stable constantmean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Yeas perturbations of large centered coordinate spheres minimize area among allcompeting surfaces that enclose the same volume. This confirms a conjecture ofH. Bray. Our results are consistent with the uniqueness results forvolume-preserving stable constant mean curvature surfaces in initial data setsobtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian.The additional hypotheses that the surfaces be spherical and far out in theasymptotic region in their results are not necessary in our work.

Large isoperimetric surfaces in initial data sets