In 2026, Peter Hintz proved the nonlinear stability of the full subextremal Kerr family (∣a∣<m|a|<m) as solutions of the Einstein vacuum equation, but only for initial data whose difference to Kerr data admits a finite asymptotic expansion in powers of r−1r^{-1} and log⁡r\log r (partial polyhomogeneity). The paper states that removing this structural hypothesis — proving stability for 'structureless' initial data with mere O(r−1−ϵ0)\mathcal{O}(r^{-1-\epsilon_0}) decay and no imposed expansion — cannot be done with its ideas, because the required gauge modifications become infinite-dimensional in nature. The problem is to prove nonlinear stability of subextremal Kerr black holes in the full subextremal range ∣a∣<m|a|<m for such structureless, weakly decaying initial data, retaining the physical conclusion of quantitative relaxation to a nearby Kerr spacetime.

Background

The Kerr spacetimes (Kerr 1963) are the two-parameter family of stationary rotating vacuum black hole solutions of the Einstein equation Ric(g)=0{\rm Ric}(g)=0, labeled by mass mm and specific angular momentum aa; they are called subextremal when ∣a∣<m|a|<m (a horizon exists) and extremal when ∣a∣=m|a|=m. The Kerr stability conjecture asserts that initial data for the Einstein vacuum equation (a Riemannian metric γ\gamma and a symmetric tensor kk on a spacelike hypersurface, satisfying the constraint equations that arise from the Gauss–Codazzi decomposition of Ric(g)=0{\rm Ric}(g)=0) which are close to the data of a subextremal Kerr spacetime and asymptotically flat — γ−γb0\gamma-\gamma_{b_0} decaying like O(r−1−ϵ0)\mathcal{O}(r^{-1-\epsilon_0}) as the Euclidean radius r→∞r\to\infty — evolve into a spacetime that settles down, at quantitative inverse-power rates, to a nearby member of the Kerr family.

The nonlinear stability program progressed through: polarized axisymmetric perturbations of Schwarzschild (a=0a=0) by Klainerman–Szeftel (2021); general perturbations of Schwarzschild by Dafermos–Holzegel–Rodnianski–Taylor (2021), whose theorem covers data with 'structureless' (no expansion structure) decay O(r−2−ϵ0)\mathcal{O}(r^{-2-\epsilon_0}); slowly rotating Kerr (∣a∣/m|a|/m small) by Klainerman–Szeftel (2023) and Giorgi–Klainerman–Szeftel (2024), with structureless decay O(r−5/2−ϵ0)\mathcal{O}(r^{-5/2-\epsilon_0}); and, in 2026, Hintz settled the full subextremal range ∣a∣<m|a|<m, building on linear stability of Kerr in the full subextremal range by Häfner–Hintz–Vasy (2025) and two companion papers (Hintz 2026a on constraint damping — a mechanism stabilizing the propagation of the constraint-violating components in a gauge-fixed formulation — and Hintz 2026b on tame estimates for wave-type equations). Hintz's spacetimes decay to a final Kerr metric at rate O(t~−2−ϵK)\mathcal{O}(\tilde t^{-2-\epsilon_{\mathcal K}}) in spatially compact regions, and the result is strong enough to imply the C0C^0-stability of the Kerr Cauchy horizon (the null hypersurface forming the inner boundary of the rotating black hole region, where predictability breaks down) via Dafermos–Luk (2025).

The 2026 theorem, however, imposes a structural hypothesis on the data: the differences γ~=γ−γb0\tilde\gamma=\gamma-\gamma_{b_0} and k~=k−kb0\tilde k=k-k_{b_0} must be partially polyhomogeneous, i.e. admit a finite expansion ∑(z,m)∈E0r−z(log⁡r)mc(z,m)(ω)\sum_{(z,m)\in E_0} r^{-z}(\log r)^m c^{(z,m)}(\omega) (with E0E_0 a finite subset of [1+ϵ0,3]×N0[1+\epsilon_0,3]\times\mathbb N_0 and ω\omega the angular variable) plus a remainder decaying two powers faster than r−1−ϵ0r^{-1-\epsilon_0}, namely O(r−3−ϵ0)\mathcal{O}(r^{-3-\epsilon_0}). For genuinely structureless data (a remainder controlled only in weighted Sobolev spaces, with no expansion), the theorem instead requires the much faster decay O(r−3−ϵ0)\mathcal{O}(r^{-3-\epsilon_0}). The expansion structure is not cosmetic: in Hintz's generalized wave map gauge (a harmonic-type coordinate gauge in which the Einstein equations become a quasilinear wave system, solved by a Nash–Moser iteration — a Newton-type scheme tolerating derivative loss), the asymptotic terms of the data are eliminated by gauge source terms lying in a finite-dimensional space, and invertibility of small perturbations of finite-dimensional maps is used. As explained in Remark 1.12 of the paper, structureless O(r−1−δ)\mathcal{O}(r^{-1-\delta}) data would, already in linearized theory, generate structureless O(t∗−δ)\mathcal{O}(t_*^{-\delta}) center-of-mass motions (pure gauge translations and boosts of the final black hole, parameterized by an infinite-dimensional function space), which no finite-dimensional gauge modification can absorb; the relevant linear maps invert only with a loss of derivatives, and perturbative invertibility fails in such topologies. This is the black-hole analogue of the classical boost problem (the failure of the center-of-mass and boost to be well-defined for data decaying as slowly as r−1r^{-1}).

The gap is exactly where physically motivated data live: Kehrberger's 'case against smooth null infinity' series and the scattering theory of Kadar–Kehrberger (2025) for quasilinear waves show that data forced by gravitational scattering are polyhomogeneous with index sets containing r−1r^{-1}-terms — slightly too weak for the 2026 theorem, whose hypotheses demand the expansion structure. Fang–Szeftel–Touati (2025) proved that constraint-satisfying initial data with any prescribed inverse-polynomial decay (no faster than prescribed) exist in abundance, so the open question concerns evolution, not data availability. Polyhomogeneity of radiative asymptotics was similarly decisive in the Minkowski-space stability theorem of Hintz–Vasy (2020).

Problem Statement

Prove the following type of theorem. Let b0=(m0,a0)b_0=(m_0,a_0) be subextremal Kerr parameters, ∣a0∣<m0|a_0|<m_0, i.e. the full subextremal range, as in the 2026 theorem; restricting to axisymmetry, to Schwarzschild, or to small angular momenta is not expected to simplify the argument, since the source's approach treats the full subextremal range of angular momenta uniformly (Remark 1.4 of arXiv:2606.28253). Suppose (γ,k)(\gamma,k) are smooth initial data on an asymptotically Euclidean hypersurface satisfying the Einstein vacuum constraint equations, with γ−γb0\gamma-\gamma_{b_0} and k−kb0k-k_{b_0} small and decaying at the weak level O(r−1−ϵ0)\mathcal{O}(r^{-1-\epsilon_0}) and O(r−2−ϵ0)\mathcal{O}(r^{-2-\epsilon_0}) respectively for some ϵ0>0\epsilon_0>0, measured in weighted Sobolev norms — analogous to the source's bb-Sobolev remainder norms, but without any expansion terms — that are weak enough to admit data with exactly these leading decays and no finite asymptotic expansion, and that do not effectively impose the O(r−3−ϵ0)\mathcal{O}(r^{-3-\epsilon_0}) decay or expansion structure already covered by the 2026 theorem. That is, the data are required to be structureless: no finite asymptotic expansion in powers of r−1r^{-1} and log⁡r\log r may be assumed. Then the maximal Cauchy development should contain an exterior region in which the solution of Ric(g)=0{\rm Ric}(g)=0 settles down to a nearby subextremal Kerr spacetime with final parameters bfb_f (well-defined and subextremal), with quantitative inverse-power decay of the same strength as the 2026 theorem — e.g. O(t~−2−ϵK)\mathcal{O}(\tilde t^{-2-\epsilon_{\mathcal K}}) in spatially compact regions, where t~\tilde t is the time function on the Kerr background equal to Boyer–Lindquist time for large rr (equivalently, O(t∗−2−ϵK)\mathcal{O}(t_*^{-2-\epsilon_{\mathcal K}}) in spatially compact subsets of {r≥m0}\{r\geq m_0\} in the source's hyperboloidal time t∗t_*) — together with quantitative decay in all asymptotic regions, in particular O(r−1)\mathcal{O}(r^{-1}) decay near null infinity with an existing radiation field, where polyhomogeneous (rather than smooth) structure at null infinity is permitted. The formulation of gauge is part of the problem: the solution may use any gauge treatment, provided the conclusion is gauge-invariant (decay of the metric to a Kerr metric up to diffeomorphism).

References

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