Inventiones mathematicae

, Volume 172, Issue 3, pp 459–475 | Cite as

On the capacity of surfaces in manifolds with nonnegative scalar curvature

  • Hubert Bray
  • Pengzi Miao


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.


Black Hole Scalar Curvature Positive Mass Theorem Round Ball Nonempty Boundary 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.School of Mathematical SciencesMonash UniversityVictoriaAustralia

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