Extremal Reissner–Nordström–Tangherlini rigidity at equality in the charged positive mass inequality for higher-dimensional black-hole initial data

For time-symmetric, source-free Einstein–Maxwell initial data that embed isometrically as graphs in Euclidean space with a mean-convex, star-shaped minimal horizon, de Lima–Girão–Lozório–Silva (Class. Quantum Grav. 33 (2016) 035008) proved the charged positive mass inequality $m\ge|Q|$ and explicitly asked whether equality $m=|Q|$ forces the data to be isometric to the extremal Reissner–Nordström–Tangherlini solution, observing rigidity indications (inverse mean curvature leaves are round spheres and satisfy the Reissner–Nordström scalar curvature identity) but not the classification. The literature status through August 2026: the inequality itself is now known well beyond the graphical class (spinorial proofs in $n\ge4$, and a conformal reduction of the $X$-ADM mass yielding the mass–charge inequality in all dimensions without spin), the complete equality classification in $3+1$ dimensions without boundary has just been achieved (equality forces an embedding into a Majumdar–Papapetrou spacetime), and extremal Reissner–Nordström rigidity in all dimensions $n\ge3$ is known under an additional Yamabe-type flux boundary condition or for asymptotically cylindrical ends. The unconditional rigidity statement for the source's graphical class with compact horizon — in any dimension $n\ge3$ — remains open and is the subject of this record.

Background

An initial data set for the Einstein–Maxwell equations (general relativity coupled to electromagnetism) in a spacetime of dimension $n+1$ consists of a Riemannian $n$-manifold $(M,g)$ carrying an electric field $E$ (in the time-symmetric case, where the extrinsic curvature of $M$ in spacetime vanishes). The constraints reduce to $R_g-(n-1)(n-2)|E|_g^2=2\mu$ with energy density $\mu\ge0$ and, for source-free Maxwell data, $\mathrm{div}_g E=0$. The ADM mass $m$ is the total gravitational mass measured at spatial infinity, defined by a limit of surface integrals over large coordinate spheres, and the total charge $Q$ is the normalized flux of $E$ through those spheres. The Reissner–Nordström–Tangherlini (RNT) solution is the spherically symmetric charged black hole of $n+1$-dimensional general relativity; it is subextremal when the mass and charge parameters satisfy $|q|<m$, and extremal when $|q|=m$, in which case its inner and outer horizons coincide into a single degenerate horizon. The charged positive mass theorem (mass–charge inequality) asserts $m\ge|Q|$ for all such data; in dimension $n=3$ it goes back to Gibbons and Hull, who proved it by Witten's spinor method (a supergravity Bogomolny-type bound), and it was conjectured in all dimensions at the time of the source paper. The equality case is expected to be rigid: $m=|Q|$ should force the data to be the exterior of an extremal RNT black hole — or, when several horizons or ends are allowed, an extremal electrostatic Majumdar–Papapetrou multi-black-hole configuration (a static superposition of extremal black holes held in equilibrium by electric repulsion). The charged Penrose inequality refines this by bounding $m$ below by the RNT expression in the horizon area radius and charge, with equality exactly for RNT; proposed by Jang in $n=3$ as a test of the cosmic censorship conjecture, it remains open in general (in $n=3$ even the inequality fails for disconnected horizons, by a Majumdar–Papapetrou-based counterexample of Weinstein and Yamada). de Lima–Girão–Lozório–Silva studied data of dimension $n\ge3$ that are asymptotically flat (the metric approaches the Euclidean metric at spatial infinity with the standard decay), carry a single outermost compact minimal boundary $\Sigma_0$ (the horizon), satisfy $\mathrm{div}_g E=0$ and $R_g\ge(n-1)(n-2)|E|_g^2$, and whose underlying manifold embeds isometrically in Euclidean space $\mathbb{R}^{n+1}$ as the graph of a function over the exterior of $\Sigma_0\subset\mathbb{R}^n$, where $\Sigma_0$ is assumed mean-convex (nonnegative mean curvature $H\ge0$ with respect to the outward normal) and star-shaped (a graph over the round sphere). Using the inverse mean curvature flow — the evolution of hypersurfaces with normal speed equal to the reciprocal of their mean curvature, whose weak formulation underlies Huisken and Ilmanen's proof of the uncharged Riemannian Penrose inequality — they proved $m\ge|Q|$ for this graphical class. Their proof yields structural information at equality: the inverse mean curvature leaves are round spheres, the horizon satisfies $|Q|=2c_n\int_{\Sigma_0}H\,d\Sigma$ (with $c_n=1/(2(n-1)\omega_{n-1})$, where $\omega_{n-1}$ is the area of the unit round $(n-1)$-sphere), and the scalar curvature satisfies the RNT identity $R_g=(n-1)(n-2)|E|_g^2$ along the flow leaves. Nevertheless they could not conclude that equality implies isometry to the extremal RNT solution, and they flagged this rigidity question twice (Remarks 3.3 and 4.1, the latter ending 'We hope to address this question elsewhere'). For orientation, the analogous equality question for uncharged graphical data was settled by Huang and Wu: an asymptotically flat graph with minimal boundary and nonnegative scalar curvature attaining equality in the Penrose inequality must be a Schwarzschild graph. The paper also left open three further threads recorded below: the charged Penrose inequality beyond the graphical class, the two-convex Alexandrov–Fenchel inequality (a Minkowski-type inequality controlling the total mean curvature of a closed hypersurface that is mean-convex and has nonnegative induced scalar curvature, due to Guan and Li for star-shaped such hypersurfaces) beyond star-shaped outer-minimizing horizons, and negative-cosmological-constant counterparts.

Problem Statement

Let $n\ge3$ and let $(M,g,E)$ be a time-symmetric, source-free Einstein–Maxwell initial data set in the sense of de Lima–Girão–Lozório–Silva: $(M,g)$ is an asymptotically flat Riemannian $n$-manifold (the metric approaches the Euclidean metric at spatial infinity with the standard decay) whose boundary is a single outermost compact minimal hypersurface $\Sigma_0$ (the horizon), $E$ is a vector field with $\mathrm{div}_g E=0$ and $|E|=O(r^{-(n-1)})$, and the Hamiltonian constraint $R_g\ge(n-1)(n-2)|E|_g^2$ holds. Assume that $(M,g)$ embeds isometrically in $\mathbb{R}^{n+1}$ as the graph of a function over the exterior of $\Sigma_0\subset\mathbb{R}^n$, and that $\Sigma_0$, viewed as a hypersurface of $\mathbb{R}^n$, is mean-convex and star-shaped (hence connected) — exactly the class for which the charged positive mass inequality $m\ge|Q|$ was proved. Prove or disprove: if the ADM mass and total charge satisfy $m=|Q|\ne0$, then $(M,g,E)$ is isometric to the exterior region of the extremal Reissner–Nordström–Tangherlini solution of charge $Q$, with $\Sigma_0$ corresponding to its degenerate horizon; equivalently, no graphical initial data in this class other than the extremal Reissner–Nordström–Tangherlini exterior attain equality in the charged positive mass inequality. A positive answer without the graphical embedding — for arbitrary asymptotically flat, source-free Einstein–Maxwell data with horizon in $n\ge3$ — would close the equality case of the charged positive mass theorem in full generality. For calibration of what is already known: the boundaryless (asymptically cylindrical) case in $3+1$ dimensions is settled by the Majumdar–Papapetrou classification of Hirsch–Zhang, and all dimensions are settled by Raulot under the additional Yamabe-type flux boundary condition $H+(n-1)|g(E,\nu)|\le\mathrm{Vol}(\partial M,h)^{-1/(n-1)}\sqrt{\frac{n-1}{n-2}\mathcal{Y}(\partial M,[h])}$ on the horizon, where $\mathcal{Y}$ is the Yamabe invariant (the infimum, over conformal rescalings of the induced horizon metric $h$, of its normalized total scalar curvature); the statement above, with no flux condition and a compact horizon, is the open core.

References

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