The p-Harmonic Capacity of an Asymptotically Flat 3-Manifold with Non-negative Scalar Curvature

Abstract

As an inclusive \({(1,3)\ni p}\)—extension of Bray–Miao’s Theorem 1 and Corollary 1 (Invent Math 172:459–475, 2008) for p = 2, this note presents a sharp isoperimetric inequality for the p-harmonic capacity of a surface in the complete, smooth, asymptotically flat 3-manifold with non-negative scalar curvature, and then an optimal Riemannian Penrose type inequality linking the ADM/total mass and the p-harmonic capacity by means of the deficit of Willmore’s energy. Even in the Euclidean 3-space, the discovered result for \({p \not =2}\) is new and non-trivial.