General stationary vacuum black-hole uniqueness conjecture

PDF p. 1: §1.1, opening conjecture; PDF p. 4: Theorem 1.1; PDF p. 5: §1.3, degenerate horizons and multiple black holes

The article asks whether the domain of outer communications of every regular, four-dimensional, asymptotically flat stationary vacuum black hole is isometric to a Kerr exterior. Its opening does not give a single precise definition of regularity for this general question. In particular, the non-degenerate Definition 1.1 below is not imposed on the whole conjecture.

A precise version of the question is: for smooth, time-oriented four-dimensional vacuum data $(M,g,T)$ that are $I^{+}$-regular relative to a specified asymptotically flat end and satisfy the black-hole hypothesis specified below, must the corresponding domain of outer communications be isometric to one of the Kerr exteriors with $M_{\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{r}}>0$ and $\mathrm{\mid }a\mathrm{\mid }\le M_{\mathrm{K}\mathrm{e}\mathrm{r}\mathrm{r}}$? Here $\mathrm{Ric}(g)=0$ is the vacuum equation and $T$ is a stationary Killing field, meaning ${\mathcal{L}}_{T}g=0$ and $T$ is future timelike on the specified stationary end. The $I^{+}$-regularity conditions, including complete $T$, global hyperbolicity of the exterior and the specified horizon cross-section conditions, are exactly those of Staticity, axisymmetry and black-hole regularity hypotheses. For this precise question, define the black-hole region relative to the selected end by $\mathcal{B}:=M\setminus I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$ and require $\mathcal{B}\mathrm{\ne }\mathrm{\varnothing }$, with $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and chronology as specified below. This is an end-relative convention using chronology in $M$; a conformal completion is not part of these data. The question makes no separate connectedness, analyticity or horizon non-degeneracy assumption.

For the end-generated spacetime region $M_{\mathrm{e}\mathrm{x}\mathrm{t}}$ specified in that definition, the exterior is $I_M^{+}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})\cap I_M^{-}(M_{\mathrm{e}\mathrm{x}\mathrm{t}})$: $I_M^{+}(U)$ consists of points reached from $U$ by future-directed timelike curves and $I_M^{-}(U)$ consists of points from which such a curve reaches $U$. An isometry is a diffeomorphism pulling the Kerr metric back to $g$ on the exterior. The Kerr metric and its exterior are those of Kerr metric, parameters, exterior and extremality. The causal-set conventions are those of Causal exterior, chronological boundaries and Cauchy data. Work in ZFC, including choice, as in the cited geometric interfaces. This precise version is a formulation of the broad question, not a claim that the article equates its informal regularity with $I^{+}$-regularity.

As of the article dated 12 November 2026, the general question is presented as a conjecture. The article claims that, for the regular non-degenerate stationary vacuum data of its Definition 1.1, if $T$ is not identically zero on the bifurcation sphere and the specified portion of ${\mathrm{\Sigma }}_0$ is maximal, the exterior is isometric to a subextremal Kerr exterior with $0<\mathrm{\mid }J\mathrm{\mid }<M^2$. The precise definition is recorded in Regular non-degenerate stationary vacuum spacetime; maximal means that the mean curvature of the spacelike hypersurface vanishes, as in Spacelike sections, maximality and lapse–shift data. This is a reported conditional claim, not a proof or an assumption in the question above.

The article separately identifies uniqueness for a smooth rotating degenerate horizon without analyticity, and the existence or exclusion of multiple horizons in the smooth $I^{+}$-regular vacuum setting, as questions of interest. A degenerate Killing horizon has vanishing surface gravity in the stated horizon convention; multiple horizons mean multiple horizon components. The comparison interface specifies surface gravity and the distinction between connected and disconnected horizons. No resolution or independent acceptance of the article's claimed partial result is asserted here.

The cited article supplies attribution and its stated 2026 status only. The open question is not a logical supplier for any theorem.