Asymptotically hyperbolic Penrose inequality

About 12 years old · traced to 2014

Suppose that $(M,g)$ is a compact perturbation of Schwarzschild-anti-de Sitter of mass $\boldsymbol{m}\text{?}$ and has scalar curvature $R_g\geq -6$ . If $m_{\partial M}\geq 0$ satisfies

\mathcal H_g^2(\partial M)\geq \mathcal H_{\overline g_{m_{\partial M}}}^2(\partial\overline M_{m_{\partial M}}),

then $\boldsymbol{m}\geq m_{\partial M}$ , with equality if and only if $(M,g)$ is isometric to Schwarzschild–anti-de Sitter of mass $m$ . Asymptotically hyperbolic Penrose inequality. This asserts that the mass of the asymptotically hyperbolic manifold is bounded below by the boundary mass determined by its horizon area, with equality only for Schwarzschild–anti-de Sitter. The conjecture is presented as a conjectured inequality; the supplied text gives no resolution status.

References

Primary source

Otis Chodosh, “Large isoperimetric regions in asymptotically hyperbolic manifolds”, arXiv:1403.6108 (2014).

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RemarkAI-assistedClaimed by OpenAI. Claims the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. The hyperbolic metric four-flux is assumed to be future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts.See full solutionHide full solution

Claimed by OpenAI. Claims the spacetime Penrose Inequality for smooth three-dimensional initial-data exteriors with one spherical asymptotically anti-de Sitter end, satisfying the stated decay and integrability assumptions, the dominant energy condition, and a compact weakly future outer trapped boundary. The hyperbolic metric four-flux is assumed to be future timelike; its Lorentz norm is the mass, and the area is the infimum over full enclosing cuts.

Scope relative to this problem: This is related asymptotically hyperbolic spacetime progress with one spherical conformal end, future-timelike mass four-flux/covector and minimum-enclosing-area bounds under the stated decay, integrability and weak future trapping; the maximal companion assumes maximal data. The target is phrased as a Riemannian compact perturbation with physical boundary area and an unspecified mass parameter. Those definitions and target equality are not identified with the source theorem; no complete target resolution is asserted.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity-October-5-2026/paper.pdf

OpenAI-260-06-The-nonmaximal-anti-de-Sitter-Penrose-Inequality-and-original-data-rigidity.pdf1,303,894 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary.See full solutionHide full solution

Claimed by OpenAI. Claims the sharp Penrose inequality for three-dimensional maximal asymptotically hyperbolic initial data with spherical conformal infinity. Under the dominant energy condition, the stated decay and integrability assumptions, and a future-timelike mass covector, the invariant mass is bounded below by the Schwarzschild–anti-de Sitter mass associated with the minimum enclosing area of a weakly future outer-trapped boundary.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data-October-5-2026/paper.pdf

OpenAI-260-07-The-Penrose-inequality-for-maximal-asymptotically-hyperbolic-initial-data.pdf853,015 bytesOpen

RemarkAI-assistedClaimed by OpenAI. Claims the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions.See full solutionHide full solution

Claimed by OpenAI. Claims the asymptotically hyperbolic Penrose inequality for sufficiently small maximal vacuum conformal perturbations of a positive-mass Schwarzschild–anti-de Sitter exterior, for every fixed decaying transverse-traceless seed and every solution branch satisfying the stated decay and mass assumptions.

Scope relative to this problem: This is a local-family result for sufficiently small maximal vacuum conformal perturbations of positive-mass Schwarzschild-AdS, each fixed decaying transverse-traceless seed and each branch satisfying the stated mass/decay assumptions. It uses the MOTS physical area, without outermostness or outer area-minimization. It does not establish all compact perturbations or the entire equality statement of the target.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data-October-5-2026/paper.pdf

OpenAI-260-08-A-local-Penrose-inequality-for-conformal-perturbations-of-Schwarzschild-anti-de-Sitter-data.pdf551,930 bytesOpen