On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

TL;DR

This paper establishes new bounds on the capacity of surfaces in asymptotically flat 3-manifolds with nonnegative scalar curvature, linking geometric quantities like area and Willmore functional, with implications for mass inequalities.

Contribution

It provides the first upper bounds for surface capacity in such manifolds, relating it to area, Willmore functional, and mass, with equality characterizations.

Findings

Derived an upper bound for surface capacity in terms of area and Willmore functional.

Established that equality holds for spherically symmetric spheres in Schwarzschild manifolds.

Connected capacity bounds to Hawking mass and total mass of the manifold.

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 infinity. Even in the special case of Euclidean space, 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.