Outward minimizing p-capacity, horizons, and Schwarzschild rigidity
Abstract
Let (M3,g) be a complete Riemannian manifold diffeomorphic to R3∖{0}, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass m+. For each p∈(1,3), define cO,p as the infimum of the Schwarzschild-normalized p-capacity over outward-minimizing finite-perimeter boundaries separating the two ends. We prove that m+≥ cO,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-≥ m+, with equality precisely for the two-sided spatial Schwarzschild manifold. We also show that a strict gap between cO,p and the unconstrained capacity infimum cM,p detects a horizon.
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BibTeX
@misc{park2026outward,
title = {{Outward minimizing $p$-capacity, horizons, and Schwarzschild rigidity}},
author = {Sehong Park},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09813},
eprint = {2609.09813},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09813},
}
RIS
TY - UNPB
TI - Outward minimizing p-capacity, horizons, and Schwarzschild rigidity
AU - Sehong Park
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.09813
UR - https://arxiv.org/abs/2609.09813
AB - Let (M^3,g) be a complete Riemannian manifold diffeomorphic to R^3∖{0}, with nonnegative scalar curvature. Assume that a distinguished end is asymptotically flat, with ADM mass m_+. For each p∈(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_+≥ 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_-≥ 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.
ER -