The 2013 theorem of Dain, Khuri, Weinstein, and Yamada bounds from below the area of a single axisymmetric, maximally sliced black hole in terms of ADM mass, charge, and angular momentum, with equality exactly on extreme Kerr–Newman data. This problem concerns the open questions that theorem left behind. First: does the charged lower bound A/4π≥m−m2−q2\sqrt{A/4\pi}\ge m-\sqrt{m^2-q^2} survive when the horizon area is replaced by the least area enclosing the outermost apparent horizon, and does equality then force extreme Reissner–Nordström data? Second: does the angular-momentum version extend to multiple black holes (Conjecture 3.4 of the source), whose missing ingredient is the inequality m2≥∣∑iJi∣m^2\ge|\sum_iJ_i|? The multiple-black-hole angular-momentum ingredient is not established, and the charged minimal-enclosing-area bound and its rigidity remain open.

Background

An initial data set (M,g,k,E,B)(M,g,k,E,B) for the Einstein–Maxwell equations consists of a Riemannian space (M,g)(M,g) with extrinsic curvature kk and electric and magnetic fields E,BE,B satisfying the constraint equations; in electrovacuum (no charged matter) the fields are divergence-free and the data carry a conserved total charge q2=qe2+qb2q^2=q_e^2+q_b^2. The ADM mass mm is the total mass measured at spatial infinity, and the outermost apparent horizon — the outermost marginally outer trapped surface (MOTS), the initial-data analogue of an event horizon — encloses the black hole region with area AA. Penrose's inequality bounds mm from below by horizon area; its electrovacuum refinement follows from the Kerr–Newman mass–area relation (the Christodoulou formula) m2=A16π+q22+π(q4+4J2)Am^2=\frac{A}{16\pi}+\frac{q^2}{2}+\frac{\pi(q^4+4J^2)}{A}, where JJ is the conserved angular momentum. Read as a quadratic in AA, this predicts a two-sided bound on horizon area in terms of mm, qq, and JJ — the generalized Penrose inequality. The upper bound on AA is the familiar Penrose-type statement; the lower bound, a lower bound on black hole area in terms of mass, charge, and angular momentum, is the subject of this problem.

For a single black hole, Dain–Khuri–Weinstein–Yamada (2013) proved the lower bound in the class of simply connected, axially symmetric (U(1)U(1)-invariant), maximal (Trgk=0\mathrm{Tr}_g k=0) initial data satisfying the dominant energy condition, with two ends — one asymptotically flat and the other asymptotically flat or asymptotically cylindrical — and linear momentum obeying Piηi=0P_i\eta^i=0 (no axial component; this makes the angular momentum JJ well defined and conserved, including its electromagnetic contribution): Amin⁡8π≥m2−q22−(m2−q22)2−q44−J2\frac{A_{\min}}{8\pi}\ge m^2-\frac{q^2}{2}-\sqrt{(m^2-\frac{q^2}{2})^2-\frac{q^4}{4}-J^2}, where Amin⁡A_{\min} is the least area required to enclose the horizon component (the least area of an enclosing surface, which need not be the horizon itself), with equality exactly for data induced on a slice of an extreme Kerr–Newman spacetime — the extremal solutions saturating m2=q2+q4+4J22m^2=\frac{q^2+\sqrt{q^4+4J^2}}{2}. The proof combines the area–charge inequality A≥4πq2A\ge4\pi q^2 for connected horizons, the area–angular-momentum–charge inequality for axisymmetric horizons, and the harmonic-map strategy: in Brill coordinates the mass becomes a functional of the twist potential, and m2≥∣J∣m^2\ge|J| reduces to a lower bound on the renormalized energy of singular harmonic maps into the hyperbolic plane, whose minimizers model the extreme Kerr–Newman throat (an asymptotically cylindrical end).

With more than one horizon the picture degrades. In the time-symmetric charged case, Weinstein and Yamada constructed two-horizon data (perturbing the Majumdar–Papapetrou family, the static extremal charged multi-black-hole solutions) with A/4π<m−m2−q2\sqrt{A/4\pi}<m-\sqrt{m^2-q^2}: the charged lower bound genuinely fails for multiple horizons, essentially because charge is additive while area is only subadditive. For multiple rotating black holes, the conjectured substitute (Conjecture 3.4 of the source) needs the positive-mass-type inequality m2≥∣∑iJi∣m^2\ge|\sum_iJ_i| relating ADM mass to the total angular momentum of the individual holes; that inequality, together with the established area–angular-momentum inequality A≥8π∣J∣A\ge8\pi|J|, implies the conjectured bound, though refuting the inequality would not by itself refute the bound. Proving it appears intertwined with the uniqueness theory of stationary multi-black-hole spacetimes. The charged case with J=0J=0 raises a parallel question: does the one-horizon bound A/4π≥m−m2−q2\sqrt{A/4\pi}\ge m-\sqrt{m^2-q^2} persist when AA is weakened to the least area of any surface enclosing the outermost apparent horizon — a quantity computable from initial data alone? Its missing ingredients are an area–charge inequality for the (generally non-horizon) area-minimizing enclosing surface and the equality cases of the positive mass theorem with charge (rigidity: extreme Reissner–Nordström), both still unresolved.

Problem Statement

The questions below delimit the lower-bound half of the generalized Penrose inequality beyond the proved single-black-hole, axisymmetric, maximal setting; they are the open questions posed in Dain–Khuri–Weinstein–Yamada (2013). Throughout, (M,g,k,E,B)(M,g,k,E,B) is an initial data set for the Einstein–Maxwell equations without charged matter (divgE=divgB=0\mathrm{div}_g E=\mathrm{div}_g B=0) satisfying the dominant energy condition; mm is the ADM mass, q2=qe2+qb2q^2=q_e^2+q_b^2 the squared total charge, JJ the conserved angular momentum, and Amin⁡A_{\min} the least area of a surface enclosing the object in question.

(1) Charged case with minimal enclosing area. For asymptotically flat data with one end whose outermost apparent horizon has a single component, prove or disprove

Amin⁡4π ≥ m−m2−q2,\sqrt{\frac{A_{\min}}{4\pi}}\ \ge\ m-\sqrt{m^2-q^2},

where Amin⁡A_{\min} is the least area required to enclose the outermost apparent horizon. This part comprises two independently resolvable sub-questions. (1a) The inequality itself: prove or disprove the displayed inequality; establishing (1a) resolves the inequality regardless of the equality question. (1b) The equality case, independently: does equality in (1a) force the data to arise from an extreme Reissner–Nordström spacetime (rigidity in the positive mass theorem with charge)? Settling (1b) resolves the rigidity question independently of (1a).

(2) Multiple black holes with angular momentum (Conjecture 3.4 of the source). Let (M,g,k)(M,g,k) be simply connected, axially symmetric, maximal, with one designated asymptotically flat end and finitely many other ends each asymptotically flat or asymptotically cylindrical; assume the dominant energy condition and Piηi=0P_i\eta^i=0 (the linear momentum has no component along the axial symmetry generator η\eta, which makes the individual angular momenta JiJ_i well defined and conserved, including their electromagnetic contributions). Let AiA_i denote the areas of the stable minimal surfaces enclosing the non-designated ends and JiJ_i their angular momenta, and set A=∑iAiA=\sum_iA_i and J=∑iJiJ=\sum_iJ_i. Prove or disprove

A8π ≥ m2−m4−J2,\frac{A}{8\pi}\ \ge\ m^2-\sqrt{m^4-J^2},

with equality if and only if the data arise from a slice of an extreme Kerr spacetime.

The common objective is to establish the validity regime — and the equality cases — of lower bounds on black hole area in terms of mass, charge, and angular momentum, given that the single-horizon, horizon-area case (Dain–Khuri–Weinstein–Yamada) is settled.

References

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