Mathematics – Differential Geometry
Scientific paper
2011-02-15
Mathematics
Differential Geometry
29 pages
Scientific paper
We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) with non-negative scalar curvature that are $C^0$-asymptotic to Schwarzschild with mass m>0 at infinity. We refine an argument of H. Bray's to obtain an effective volume comparison theorem 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 round, centered spheres. This implies as a corollary that the volume preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate balls minimize area among competitors enclosing the same volume. This confirms a conjecture of H. Bray's. Our results are consistent with the uniqueness results for volume preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. Their hypotheses that these surfaces be spherical and far out in the asymptotic regime are not necessary in our work. In particular, we obtain a full description of the large isoperimetric surfaces in these data sets without imposing symmetry assumptions. Our results are relevant in the context of notions of isoperimetric mass as developed by G. Huisken.
Eichmair Michael
Metzger J. J.
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