Endpoint capacity at strictly stable minimal boundaries:Gaussian layers and Schwarzschild deficits

Abstract

Let (Mⁿ,g), 3≤ n≤7, be a one-ended asymptotically flat manifold whose boundary Σ is a connected, strictly stable, outermost minimal hypersurface. As p↓1, the p-capacity of Σ converges to its area, by a theorem of Fogagnolo and Mazzieri. We establish a higher-order asymptotic expansion governed by a single function on Σ: the positive solution g of the reciprocal Jacobi equation Jg=g⁻¹. After a universal term, the first correction is determined by the average of log g, and the capacitary potential drops from one to zero across a Gaussian boundary layer of width g√(p−1); under the normalized energy measure, the logarithm of the gradient has limiting variance 1/2 plus the variance of log g. In dimension three, assuming in addition that R_g+|II|²≥0 along Σ, we identify the capacity deficit of Σ relative to the Schwarzschild horizon of the same area at order p−1: it is nonnegative, vanishes exactly when the induced metric of Σ is round and R_g+|II|²≡0, and controls the deviation of g from a constant through a relative-entropy identity.

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BibTeX RIS

BibTeX

@misc{park2026endpoint,
  title = {{Endpoint capacity at strictly stable minimal boundaries:Gaussian layers and Schwarzschild deficits}},
  author = {Sehong Park},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25444},
  eprint = {2610.25444},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25444},
}

RIS

TY  - UNPB
TI  - Endpoint capacity at strictly stable minimal boundaries:Gaussian layers and Schwarzschild deficits
AU  - Sehong Park
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25444
UR  - https://arxiv.org/abs/2610.25444
AB  - Let (M^n,g), 3≤ n≤7, be a one-ended asymptotically flat manifold whose boundary Σ is a connected, strictly stable, outermost minimal hypersurface. As p↓1, the p-capacity of Σ converges to its area, by a theorem of Fogagnolo and Mazzieri. We establish a higher-order asymptotic expansion governed by a single function on Σ: the positive solution g of the reciprocal Jacobi equation Jg=g^−1. After a universal term, the first correction is determined by the average of log g, and the capacitary potential drops from one to zero across a Gaussian boundary layer of width g√(p−1); under the normalized energy measure, the logarithm of the gradient has limiting variance 1/2 plus the variance of log g. In dimension three, assuming in addition that R_g+|II|^2≥0 along Σ, we identify the capacity deficit of Σ relative to the Schwarzschild horizon of the same area at order p−1: it is nonnegative, vanishes exactly when the induced metric of Σ is round and R_g+|II|^2≡0, and controls the deviation of g from a constant through a relative-entropy identity.
ER  -

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