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Equality and rigidity in the spacetime Penrose inequality

OpenAI · September 27, 2026

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Abstract

Equality in the spacetime Penrose inequality identifies the original initial data as a spacelike slice of a Schwarzschild–Tangherlini exterior, smoothly attached to its horizon, under the stated outermostness and decay hypotheses. We prove this in every spatial dimension n ≥ 3 for a strong-decay exterior class and for weaker decay in dimensions three and four. In dimension three the finite-component theorem forces a single spherical horizon and gives qualitative strictness for disconnected boundaries.

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Cite

@misc{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026,
  author = {{OpenAI}},
  title = {{Equality and rigidity in the spacetime Penrose inequality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Equality-and-rigidity-in-the-spacetime-Penrose-inequality-September-27-2026/paper.pdf}{OAI:Equality-and-rigidity-in-the-spacetime-Penrose-inequality-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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