Outward minimizing $p$-capacity, horizons, and Schwarzschild rigidity

Abstract

Let $(M^3,g)$ be a complete Riemannian manifold diffeomorphic to ${\mathbb{R}}^3\setminus \{0\}$, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass $m_{+}$. For each $p\in (1,3)$, define $c_{O,p}$ as the infimum of the Schwarzschild-normalized $p$-capacity over outward-minimizing finite-perimeter boundaries separating the two ends. We prove that $m_{+}\ge c_{O,p}$ whenever $A(g)>0$, where $A(g)$ is the infimum of the areas of boundaries separating the two ends. Equality at a single exponent produces a least-area horizon, forces its exterior to be the Schwarzschild exterior of mass $m_{+}$, and yields equality at every exponent. In the equality case, if the second end is also asymptotically flat, its mass satisfies $m_{-}\ge m_{+}$, with equality precisely for the two-sided spatial Schwarzschild manifold. We also show that a strict gap between $c_{O,p}$ and the unconstrained capacity infimum $c_{M,p}$ detects a horizon.