Smooth rotating degenerate-horizon uniqueness

PDF p. 5: §1.3, end of first paragraph

The article asks whether uniqueness holds for a smooth, stationary, asymptotically flat vacuum black hole with a connected rotating degenerate horizon without assuming analyticity. The intended uniqueness conclusion is that its domain of outer communications is isometric to an extremal Kerr exterior, under suitable regularity hypotheses.

One precise subcase is the following. Let $(M,g,T)$ be a smooth, time-oriented Lorentzian four-manifold with $\mathrm{Ric}(g)=0$ and a stationary Killing field $T$, and assume it is $I^{+}$-regular relative to a specified asymptotically flat spacelike end in the sense of Staticity, axisymmetry and black-hole regularity hypotheses. In particular, $T$ is complete, the domain of outer communications $D$ is globally hyperbolic, and the specified spacelike section has the end, boundary and horizon-generator properties stated in that definition. Suppose the nonempty future event horizon $H$ is connected and is a smooth Killing horizon for a specified generating Killing field $K$. Require the nonzero-generator part $H_{\ast }=\{p\in H:K_p\mathrm{\ne }0\}$ to be nonempty and dense in $H$, with $K$ tangent and normal to $H$ on $H_{\ast }$. It is rotating relative to $T$ when $T$ fails somewhere to lie in the horizon's null normal line; it is degenerate when the surface-gravity function of $K$ vanishes on its nonzero-generator part, as in Rotation, surface gravity and horizon degeneracy. Must $(D,g{\mathrm{\mid }}_D)$ be isometric to an extremal Kerr exterior from Kerr metric, parameters, exterior and extremality, with $M>0$ and $\mathrm{\mid }a\mathrm{\mid }=M$?

This subcase explicitly assumes a smooth horizon-generating Killing field. The broader article question also asks for a treatment without assuming such a field merely as a consequence of smooth stationarity; neither its existence nor global axisymmetry is asserted here.

The article dated 12 November 2026 presents the smooth, non-analytic rotating degenerate case as a question. Its §1.3 describes analytic rigidity and axisymmetric extremal uniqueness as partial results, with their own hypotheses. In particular, Chruściel–Nguyen's abstract states an analytic electrovacuum result for a connected rotating degenerate horizon; analyticity is an explicit hypothesis of that result. These comparisons do not answer the question above. The status recorded here is the article-date status, not a claim that a complete later literature search has ruled out a resolution.