The Penrose inequality for general asymptotically flat initial data sets beyond the 2-convexity condition

The Penrose conjecture (the non-time-symmetric, or spacetime, Penrose inequality) asks whether every complete, connected, asymptotically flat initial data set $(M^3,g,\mathbf{k})$ for the Einstein equations satisfying the dominant energy condition obeys $m \ge \sqrt{A/16\pi}$, where $m$ is the ADM mass and $A$ is the area of the outermost minimal-area enclosure of a past apparent horizon (or, in the generalized formulation, of the outermost generalized apparent horizon). The inequality is proved for time-symmetric data ($\mathbf{k}=0$), for spherically symmetric data, and — in recent 2026 work of Dong — for data with $\mathbf{k}$ proportional to $g$ (for each connected component of the outermost generalized apparent horizon) and for data satisfying the 2-convexity condition $(\operatorname{tr}_g \mathbf{k})g-\mathbf{k}\ge 0$ with connected outermost past apparent horizon, via the newly introduced $\sigma$-inverse mean curvature flow. The open core is the conjecture for initial data satisfying only the dominant energy condition, with no sign or algebraic restriction on the extrinsic curvature.

Background

An initial data set for the Einstein equations is a triple $(M^3,g,\mathbf{k})$: a Riemannian 3-manifold $(M,g)$ together with a symmetric 2-tensor $\mathbf{k}$, the extrinsic curvature, representing the first and second fundamental data of a spacelike hypersurface in a spacetime. The Einstein constraint equations couple them to the matter density and current, $16\pi\mu = R+(\operatorname{tr}_g\mathbf{k})^2-|\mathbf{k}|^2$ and $8\pi J=\operatorname{div}_g(\mathbf{k}-(\operatorname{tr}_g\mathbf{k})g)$, where $R$ is the scalar curvature of $g$. The dominant energy condition (DEC) is $\mu \ge |J|_g$, expressing that matter travels no faster than light. Asymptotic flatness means that outside a compact set the geometry approaches Euclidean space with suitable decay; the ADM mass $m$ is the total mass-energy read off from the $1/r$ coefficient of the metric at infinity.

A closed surface $\Sigma$ is a marginally outer trapped surface (MOTS) if its future null expansion vanishes; on the slice this is the equation $H_\Sigma = -\operatorname{tr}_\Sigma\mathbf{k}$, with $H_\Sigma$ the mean curvature and $\operatorname{tr}_\Sigma\mathbf{k}$ the trace of $\mathbf{k}$ along the surface — a future apparent horizon. Reversing the sign gives a past apparent horizon, $H_\Sigma = \operatorname{tr}_\Sigma\mathbf{k}$. A generalized apparent horizon satisfies $H_\Sigma = |\operatorname{tr}_\Sigma\mathbf{k}|$, i.e. is a horizon of either orientation; Eichmair (2010) proved existence and regularity of an outermost such horizon. The relevant area $A$ is that of the outermost minimal-area enclosure of the horizon: the least-area surface (allowing non-connected competitors) enclosing it.

Penrose's 1973 heuristic, combining cosmic censorship — the expectation that gravitational collapse never leaves singularities visible to distant observers, only ones hidden inside black holes — with Hawking's area theorem — the result that the total event-horizon area of a black hole never decreases forward in time — predicts $m \ge \sqrt{A/16\pi}$. In the time-symmetric case $\mathbf{k}=0$ horizons are minimal surfaces and the statement becomes the Riemannian Penrose inequality, proved by Huisken and Ilmanen (2001) via inverse mean curvature flow (IMCF — the flow of surfaces outward with normal speed equal to the reciprocal $1/H$ of their mean curvature) for connected horizons, and by Bray (2001) via a conformal flow (a deformation of the metric by pointwise positive rescalings, arranged so that the ratio of mass to horizon area decreases toward its extreme value) in full generality. For general $\mathbf{k}$ the conjecture is open. Partial cases: spherical symmetry (Hayward 1996; Iriondo–Malec–Ó Murchadha 1996; Bray–Khuri 2010); a Jang-equation reduction (the Jang equation is a prescribed-mean-curvature equation coupling $g$ and $\mathbf{k}$ whose solutions realize the slice as a graph in an auxiliary spacetime) of the general conjecture to an unresolved PDE system (Bray–Khuri 2011); inverse spacetime mean curvature flow (a spacetime analogue of IMCF driven by the reciprocal of the surface's spacetime mean curvature, i.e. the norm of its mean-curvature vector in the ambient spacetime) in the maximal case $\operatorname{tr}_g\mathbf{k}=0$ (slices with vanishing mean curvature) (Huisken–Wolff 2025); mass lower bounds of Penrose type with a suboptimal universal constant (Allen–Bryden–Kazaras–Khuri 2025); dynamical versions inside Kerr black-hole formation spacetimes (Kerr being the stationary rotating vacuum black holes; An–He 2025); and conditional results under a quasi-final-state hypothesis (a precise late-time decay condition on the apparent horizon; Ellithy 2026).

The source paper (Dong, arXiv:2607.20741) proved the conjecture for a further class: defining $\mathbf{P}=(\operatorname{tr}_g\mathbf{k})g-\mathbf{k}$, the 2-convexity condition — the sum of the two smallest eigenvalues of $\mathbf{k}$ being nonnegative — is equivalent to $\mathbf{P}\ge 0$, and under DEC plus $\mathbf{P}\ge 0$ with a connected outermost past apparent horizon, Dong proves $m \ge \sqrt{|\Sigma|/16\pi}$ with equality only for the Schwarzschild slice with $\mathbf{k}\equiv 0$ (the time-symmetric exterior region of the unique spherically symmetric vacuum black hole). The tool is the $\sigma$-IMCF introduced in Dong's companion paper (arXiv:2605.26504): the flow $\partial F/\partial t = \nu/(H-\sigma(\nu,\nu))$ with $\sigma=\mathbf{P}$, along which a Geroch-type monotone quantity $m_{\mathbf{P}}$ — the analogue, for this flow, of the Hawking quasilocal mass (a standard functional of a surface's area and mean curvature) whose IMCF-monotonicity powers the Riemannian proof — is monotone under DEC. The monotonicity closing argument fails at exactly one point: without $\mathbf{P}\ge 0$ (a pointwise algebraic condition on $\mathbf{k}$) no known flow or monotone quantity plays the role that IMCF plays for $\mathbf{k}=0$.

Problem Statement

Prove or disprove the Penrose conjecture for general initial data sets (Conjecture 1.1 of Dong, arXiv:2607.20741 — the standard non-time-symmetric formulation): let $(M^3,g,\mathbf{k})$ be a smooth, complete, connected, asymptotically flat initial data set satisfying the dominant energy condition $\mu \ge |J|_g$, with ADM mass $m$. If $\Sigma \subset M$ is a past apparent horizon ($H_\Sigma=\operatorname{tr}_\Sigma\mathbf{k}$), prove that

$$m \ge \sqrt{\frac{A}{16\pi}}.$$

Here $A$ is the area of the outermost minimal-area enclosure of $\Sigma$. The same question may be posed with $A$ the area of the outermost generalized apparent horizon ($H_\Sigma=|\operatorname{tr}_\Sigma\mathbf{k}|$) — the generalized version; the two formulations are known to coincide under the 2-convexity condition (Remark 1.5 of the source paper), and both are open in general. The open core is the case of no restriction on the extrinsic curvature $\mathbf{k}$ beyond the DEC — in particular, data not satisfying the 2-convexity condition $(\operatorname{tr}_g\mathbf{k})g-\mathbf{k}\ge 0$, not satisfying $\mathbf{k}$ proportional to $g$, and not covered by time symmetry or spherical symmetry. Within the 2-convexity class, the source's Theorem 1.2 also proves the rigidity assertion that equality forces $\mathbf{k}\equiv 0$ and $(M,g)$ isometric to a Schwarzschild slice, but it assumes a connected outermost past apparent horizon; the disconnected-horizon (multiple black holes) case is not covered even by that theorem, although Conjecture 1.1 ranges over arbitrary past apparent horizons.

References

1. Conghan Dong, The Penrose conjecture for initial data sets satisfying a 2-convexity condition, 2026, arXiv:2607.20741, DOI 10.48550/arXiv.2607.20741, https://arxiv.org/abs/2607.20741
2. Conghan Dong, The σ-inverse mean curvature flow and the generalized Penrose conjecture, 2026, arXiv:2605.26504, DOI 10.48550/arXiv.2605.26504, https://arxiv.org/abs/2605.26504
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