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On the capacity of surfaces in manifolds with nonnegative scalar curvature

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Abstract

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at ∞. Even in the special case of ℝ3, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

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Authors and Affiliations

  1. Department of Mathematics, Duke University, Durham, NC, 27708, USA

    Hubert Bray

  2. School of Mathematical Sciences, Monash University, Victoria, 3800, Australia

    Pengzi Miao

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  1. Hubert Bray
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  2. Pengzi Miao
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Correspondence to Pengzi Miao.

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Bray, H., Miao, P. On the capacity of surfaces in manifolds with nonnegative scalar curvature . Invent. math. 172, 459–475 (2008). https://doi.org/10.1007/s00222-007-0102-x

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  • DOI: https://doi.org/10.1007/s00222-007-0102-x

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