Bray, H; Miao, P, *On the capacity of surfaces in manifolds with nonnegative scalar curvature*,
Inventiones Mathematicae, vol. 172 no. 3
(June, 2008),
pp. 459-475, Springer Nature [arXiv:0707.3337v1], [doi] .
**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. © 2008 Springer-Verlag.*