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Conformal flow and the Riemannian Penrose inequality with minimizing frontiers

OpenAI · September 27, 2026

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Abstract

We give a detailed conformal-flow proof of the numerical Riemannian Penrose inequality in every dimension n ≥ 3 for complete asymptotically flat exteriors with nonnegative scalar curvature and full compact frontiers that are outer minimizing and locally perimeter minimizing. The metric extends smoothly through the possibly singular frontier; neither spin nor frontier connectedness is assumed. The proof follows the conformal-flow approach of Bray, Bray–Lee, and Bi–Zhu.

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Cite

@misc{OAI:Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026,
  author = {{OpenAI}},
  title = {{Conformal flow and the Riemannian Penrose inequality with minimizing frontiers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026/paper.pdf}{OAI:Conformal-flow-and-the-Riemannian-Penrose-inequality-with-minimizing-frontiers-September-27-2026}},
  year = {2026}
}

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Family 260: Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.

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