The sharp Penrose inequality for asymptotically (locally) hyperbolic initial data

For time-symmetric initial data with negative cosmological constant — complete Riemannian manifolds that are asymptotically hyperbolic (AH) or asymptotically locally hyperbolic (ALH) with scalar curvature at least $-n(n-1)$ and an outermost minimal (horizon) boundary $\Sigma$ — the conjectured sharp Penrose inequality asserts that the Chruściel–Herzlich mass is bounded below by the explicit Kottler mass–area function of $|\Sigma|$, with equality exactly for the Kottler (topological Schwarzschild–anti-de-Sitter) black hole metrics. As formulated in the primary sources (Gibbons; Chruściel–Simon; de Lima–Girão; Ge–Wang–Wu–Xia), the conjecture is open in this generality: it is proven only for graphical data, for nonpositive mass, for small perturbations of Schwarzschild–AdS, and under spherical or cohomogeneity-one symmetry; the best unconditional bounds known carry a universal but vastly suboptimal constant. The companion non-time-symmetric (spacetime) inequality for asymptotically hyperboloidal data satisfying the dominant energy condition is a sibling of, rather than identical to, the ALH conjecture and is known in its sharp form only conditionally, via solvability of a coupled Jang–inverse-mean-curvature-flow system; and below the genus-dependent area threshold $|\Sigma|=\vartheta_{n-1}$ for hyperbolic conformal infinity — where the conjectured bound is no longer positive and static metrics already obey a reverse inequality — the sharp mass–area relation is itself unsettled. These companion questions are context for, not part of, the conjecture stated below.

Background

An initial data set $(M,g,k)$ for the Einstein equations consists of a Riemannian metric $g$ and a symmetric 2-tensor $k$ (the second fundamental form of the slice in spacetime), together with the energy density $\mu$ and momentum density $J$ fixed by the constraint equations. The dominant energy condition (DEC) requires $\mu\ge|J|_g$; in the time-symmetric case $k=0$ it reduces to the scalar curvature bound $R_g\ge -n(n-1)$ when the cosmological constant is $\Lambda=-n(n-1)/2<0$. A closed 2-sided hypersurface is a marginally outer trapped surface (MOTS), informally an apparent horizon, when its future null expansion vanishes; when $k=0$ this is exactly a minimal surface. The Riemannian Penrose conjecture predicts that the total mass of a slice bounds the area $|\Sigma|$ of its outermost horizon from below; in the asymptotically flat case it was proved by Huisken–Ilmanen (single horizon component, via inverse mean curvature flow, IMCF — the flow moving hypersurfaces outward with speed $1/H$, $H$ being the mean curvature, which monotonizes the Hawking quasi-local mass) and by Bray (general case, via a conformal flow of metrics).

With negative cosmological constant the model geometries are the Kottler (topological Schwarzschild–AdS) metrics $g_{\epsilon,m}=dr^2/\rho_{\epsilon,m}(r)^2+r^2h$ on $(r_{\epsilon,m},\infty)\times N$, where $(N,h)$ is a closed space form of sectional curvature $\epsilon$ ($\epsilon=1$ spherical, giving the asymptotically hyperbolic (AH) case; $\epsilon=0$ toroidal; $\epsilon=-1$ hyperbolic, i.e. conformal infinity of genus $\gamma\ge 2$ in dimension $n=3$), $\rho_{\epsilon,m}^2=r^2+\epsilon-2m/r^{n-2}$, and $N$ has area $\vartheta_{n-1}$. A complete manifold whose end is asymptotic to the reference metric $g_\epsilon=dr^2/(r^2+\epsilon)+r^2h$ — in the sense that, in a chart $\Psi$ at conformal infinity, $\|\Psi_*g-g_\epsilon\|_{g_\epsilon}+\|d\Psi_*g\|_{g_\epsilon}=O(r^{-\sigma})$ for some $\sigma>n/2$, with $R_g+n(n-1)$ integrable — is asymptotically locally hyperbolic (ALH), and it carries a chart-invariant mass $m_{(M,g)}$, defined by Chruściel, Herzlich and Nagy as a boundary integral at conformal infinity. The Kottler metric $g_{\epsilon,m}$ has mass $m$ and horizon area $|H_{\epsilon,m}|$ related by $m=\tfrac12\left(\left(|H_{\epsilon,m}|/\vartheta_{n-1}\right)^{n/(n-1)}+\epsilon\left(|H_{\epsilon,m}|/\vartheta_{n-1}\right)^{(n-2)/(n-1)}\right)$. Motivated by the thermodynamics of gravitational collapse, Gibbons proposed, and Chruściel–Simon formulated in connection with static classification, the conjecture that Kottler metrics minimize mass for given horizon area: if $(M^n,g)$, $n\ge 3$, is ALH with $R_g\ge -n(n-1)$ and carries an outermost minimal horizon $\Sigma$, then $m_{(M,g)}\;\ge\;\frac{1}{2}\left(\left(\frac{|\Sigma|}{\vartheta_{n-1}}\right)^{\frac{n}{n-1}}+\epsilon\left(\frac{|\Sigma|}{\vartheta_{n-1}}\right)^{\frac{n-2}{n-1}}\right)$ with equality only for the Kottler metric. For $\epsilon=-1$ the right-hand side is positive — so that the lower bound is effective — only when $|\Sigma|>\vartheta_{n-1}$, and the conjecture is asserted in that effective regime; below that genus-dependent threshold the bound is negative, and since static metrics there already obey a reverse inequality, the sharp sub-threshold mass–area relation is a companion open question. For genus $\gamma\ge1$ horizons in $n=3$ the inequality is precisely the Gibbons–Chruściel–Simon question about Penrose-type inequalities for higher-genus (topological) black holes. The naive IMCF attack fails: Neves showed that on asymptotically hyperbolic manifolds the weak IMCF does not converge suitably at infinity, so Geroch–Gibbons Hawking-mass monotonicity cannot close the argument. What is known: sharp inequalities for ALH data realized as graphs in a static locally hyperbolic ambient space (de Lima–Girão; Ge–Wang–Wu–Xia for graphs over Kottler space; earlier AH-graph results of Dahl–Gicquaud–Sakovich), for ALH manifolds with nonpositive mass aspect (Lee–Neves, via IMCF — a class complementary to the graphical one, where the mass is necessarily positive), for small perturbations of Schwarzschild–AdS (Ambrozio; Khuri–Kopiński), under spherical symmetry (Husain–Singh) and cohomogeneity-one symmetry (Khuri–Kunduri), and, in dimension $n=3$ and for asymptotically hyperbolic (spherical-infinity) Riemannian manifolds only, the suboptimal-constant corollary of the 2025 Allen–Bryden–Kazaras–Khuri spacetime result via harmonic level sets and the Jang equation.

The non-time-symmetric version concerns asymptotically hyperboloidal data $(M,g,k)$ (slices asymptotic to null infinity in an asymptotically flat spacetime, with Wang's mass $m$) satisfying the DEC; the conjecture $m\ge\sqrt{A/16\pi}$, with $A$ the minimal area enclosing the outermost apparent horizon, was proposed by X. Wang and by Mars. The generalized Jang equation $\left(g^{ij}-\frac{u^2f^if^j}{1+u^2|\nabla f|_g^2}\right)\left(\frac{u\nabla_{ij}f+u_if_j+u_jf_i}{\sqrt{1+u^2|\nabla f|_g^2}}-k_{ij}\right)=0$ (the equation for a graph in the warped product $(M\times\mathbb{R},g+u^2dt^2)$ whose scalar curvature gains a sign) reduces the spacetime inequality to geometry on the Jang surface; Cha–Khuri–Sakovich coupled it to IMCF and proved that the Penrose inequality follows if this coupled Jang–IMCF system admits a solution with prescribed hyperboloidal asymptotics — a solvability hypothesis that remains open for the sharp constant. Allen–Bryden–Kazaras–Khuri (2025) proved a bound of this shape unconditionally in the asymptotically flat and asymptotically hyperboloidal regimes, with a universal constant far from optimal.

Problem Statement

Prove or disprove the sharp Penrose inequality for time-symmetric asymptotically (locally) hyperbolic initial data at its natural level of generality — no graphical hypothesis, no symmetry assumption, and no restriction on the sign of the mass or mass aspect:

Let $(M^n,g)$, $n\ge 3$, be a complete asymptotically locally hyperbolic manifold with respect to the reference metric $g_\epsilon=dr^2/(r^2+\epsilon)+r^2h$, where $(N^{n-1},h)$ is a closed space form of sectional curvature $\epsilon$ and area $\vartheta_{n-1}$ ($\epsilon\in\{-1,0,1\}$; the spherical case $\epsilon=1$ with $(N,h)$ the round sphere is the AH case, and all three values of $\epsilon$ are required), whose end satisfies $\|\Psi_*g-g_\epsilon\|_{g_\epsilon}+\|d\Psi_*g\|_{g_\epsilon}=O(r^{-\sigma})$ for some $\sigma>n/2$ with $R_g+n(n-1)$ integrable, with $R_g\ge -n(n-1)$ and a compact outermost minimal horizon $\Sigma$ (for $\epsilon=-1$ the inequality and its equality case are asserted only in the effective regime $|\Sigma|>\vartheta_{n-1}$; the sub-threshold regime $|\Sigma|\le\vartheta_{n-1}$ is not covered by the conjecture). Establish or refute $m_{(M,g)}\;\ge\;\frac{1}{2}\left(\left(\frac{|\Sigma|}{\vartheta_{n-1}}\right)^{\frac{n}{n-1}}+\epsilon\left(\frac{|\Sigma|}{\vartheta_{n-1}}\right)^{\frac{n-2}{n-1}}\right),$ where $m_{(M,g)}$ is the Chruściel–Herzlich–Nagy mass and $\vartheta_{n-1}$ the area of the conformal infinity; characterize the equality case (conjecturally: equality holds if and only if $(M,g)$ is isometric to the corresponding Kottler metric $g_{\epsilon,m}$).

Related open questions that frame the conjecture but are not part of it: determining the correct sharp mass–area relation in the sub-threshold regime $|\Sigma|\le\vartheta_{n-1}$ for $\epsilon=-1$, where the conjectured bound is negative and the static class obeys a reverse inequality (Harvie–Wang); bridging the complementary nonpositive-mass (Lee–Neves) and positive-mass graphical regimes under a single argument; and the sibling spacetime conjecture $m\ge\sqrt{A/16\pi}$ for 3-dimensional asymptotically hyperboloidal data satisfying the dominant energy condition with connected outermost apparent horizon, which Cha–Khuri–Sakovich reduced, conditionally, to the open solvability of their coupled Jang–IMCF system with prescribed hyperboloidal asymptotics.

References

1. Levi Lopes de Lima and Frederico Girão, "A Penrose inequality for asymptotically locally hyperbolic graphs", arXiv:1304.7887 (2013), https://arxiv.org/abs/1304.7887; superseded by the published version of arXiv:1209.0438 (entry 2)
2. Levi Lopes de Lima and Frederico Girão, "An Alexandrov–Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality", Ann. Henri Poincaré 17 (2016), 979–1002, DOI 10.1007/s00023-015-0414-0, https://arxiv.org/abs/1209.0438
3. Ye Sle Cha, Marcus Khuri, Anna Sakovich, "Reduction arguments for geometric inequalities associated with asymptotically hyperboloidal slices", Classical and Quantum Gravity 33 (2016), 035009, DOI 10.1088/0264-9381/33/3/035009, https://arxiv.org/abs/1509.06255
4. Piotr T. Chruściel and Walter Simon, "Towards the classification of static vacuum spacetimes with negative cosmological constant", Journal of Mathematical Physics 42 (2001), 1779–1817, DOI 10.1063/1.1340869, https://arxiv.org/abs/gr-qc/0004032
5. G. W. Gibbons, "Some comments on gravitational entropy and the inverse mean curvature flow", Classical and Quantum Gravity 16 (1999), 1677–1687, DOI 10.1088/0264-9381/16/6/302
6. Dan A. Lee and André Neves, "The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass", Communications in Mathematical Physics 339 (2015), 327–352, DOI 10.1007/s00220-015-2421-x, https://arxiv.org/abs/1310.3002
7. Yuxin Ge, Guofang Wang, Jie Wu, Chao Xia, "A Penrose inequality for graphs over Kottler space", Calculus of Variations and Partial Differential Equations 52 (2015), 755–782, DOI 10.1007/s00526-014-0732-y, https://arxiv.org/abs/1309.6248
8. Mattias Dahl, Romain Gicquaud, Anna Sakovich, "Penrose type inequalities for asymptotically hyperbolic graphs", Ann. Henri Poincaré 14 (2013), 1135–1168, arXiv:1201.3321, https://arxiv.org/abs/1201.3321
9. André Neves, "Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds", Journal of Differential Geometry 84 (2010), 191–229, DOI 10.4310/jdg/1271271798, https://arxiv.org/abs/0711.4335
10. Levi Lopes de Lima and Frederico Girão, "Positive mass and Penrose type inequalities for asymptotically hyperbolic hypersurfaces", General Relativity and Gravitation 47 (2015), Art. 23, arXiv:1201.4991, https://arxiv.org/abs/1201.4991
11. Lucas C. Ambrozio, "On perturbations of the Schwarzschild anti-de Sitter spaces of positive mass", Communications in Mathematical Physics 337 (2015), 767–783, DOI 10.1007/s00220-015-2360-6, https://arxiv.org/abs/1402.4317
12. Marcus Khuri and Jarosław Kopiński, "Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS", Classical and Quantum Gravity 40 (2023), 045007, DOI 10.1088/1361-6382/acb24b, https://arxiv.org/abs/2209.01234
13. Brian Allen, Edward Bryden, Demetre Kazaras, Marcus Khuri, "Proof of the spacetime Penrose inequality with suboptimal constant in the asymptotically flat and asymptotically hyperboloidal regimes", arXiv:2504.10641 (2025), https://arxiv.org/abs/2504.10641
14. Brian Harvie and Ye-Kai Wang, "On the geometry and uniqueness of asymptotically locally hyperbolic static vacuum black holes", Communications in Mathematical Physics (2026), DOI 10.1007/s00220-026-05717-y, https://arxiv.org/abs/2509.18026
15. Marcus Khuri and Hari Kunduri, "The spacetime Penrose inequality for cohomogeneity one initial data", arXiv:2404.13247 (2024), Adv. Theor. Math. Phys. (to appear), https://arxiv.org/abs/2404.13247
16. Viqar Husain and Suprit Singh, "On the Penrose inequality in anti-de Sitter space", Physical Review D 96 (2017), 104055, DOI 10.1103/PhysRevD.96.104055, https://arxiv.org/abs/1709.02395
17. Ye Sle Cha and Marcus A. Khuri, "Transformations of asymptotically AdS hyperbolic initial data and associated geometric inequalities", General Relativity and Gravitation 50 (2018), no. 1, Art. 3, arXiv:1707.09398, https://arxiv.org/abs/1707.09398
18. Jaroslaw S. Jaracz, "Nonexistence of solutions to the coupled generalized Jang equation/zero divergence system", Classical and Quantum Gravity 40 (2023), 105001, DOI 10.1088/1361-6382/acf17f, https://arxiv.org/abs/2304.09332
19. Xiaodong Wang, "The mass of asymptotically hyperbolic manifolds", Journal of Differential Geometry 57 (2001), 273–299, DOI 10.4310/jdg/1090348112
20. Gerhard Huisken and Tom Ilmanen, "The inverse mean curvature flow and the Riemannian Penrose inequality", Journal of Differential Geometry 59 (2001), 353–437, DOI 10.4310/jdg/1090349447
21. Hubert L. Bray, "Proof of the Riemannian Penrose inequality using the positive mass theorem", Journal of Differential Geometry 59 (2001), 177–267, DOI 10.4310/jdg/1090349428
22. Lan-Hsuan Huang and Hyun Chul Jang, "Scalar curvature deformation and mass rigidity for asymptotically locally hyperbolic manifolds with boundary", arXiv:2108.12887 (2021), Trans. Amer. Math. Soc. (to appear), https://arxiv.org/abs/2108.12887
23. Piotr T. Chruściel and Gregory J. Galloway, "Positive mass theorems for asymptotically hyperbolic Riemannian manifolds with boundary", arXiv:2107.05603 (2021), https://arxiv.org/abs/2107.05603