Extend the noncompact proof to fully general asymptotics

Extend the proof excluding multiple inner Killing horizons in stationary-axisymmetric black holes with noncompact planar or hyperbolic horizon sections to fully general asymptotic geometries, without assuming asymptotically planar or hyperbolic symmetry along the noncompact directions.

Background

The noncompact-horizon argument uses an inverse lapse flow and a monotone flux function F(z). To establish monotonicity, the proof assumes that deviations from planar or hyperbolic symmetry are confined to compact regions, ensuring that the averaged scalar curvature of the noncompact leaves is nonpositive. The authors note that this asymptotic assumption is needed for the proof in dimensions d>3.

The unresolved problem is to remove this additional asymptotic-symmetry assumption and establish the same exclusion of noncompact stationary pockets under fully general asymptotics. The authors state that no physical counterexample is currently known, but the mathematical extension remains open.

References

Our proof for noncompact case needs an additional asymptotic symmetry when $d>3$. Extending the noncompact proof to fully general asymptotics remains a mathematically open question, though no physical counterexample is currently known.

— At Most One Inner Killing Horizon in Stationary Black Holes  (2609.16698 - Ye et al., 15 Sep 2026) in Discussion

Finally, the remaining single inner Killing horizon—when present—constitutes a Cauchy horizon. Our theorem shows that the energy conditions alone cannot eliminate it; whether additional, physically motivated restrictions on the matter could rule it out remains a decisive question, as its removal would provide a direct geometric underpinning for strong cosmic censorship.

— At Most One Inner Killing Horizon in Stationary Black Holes  (2609.16698 - Ye et al., 15 Sep 2026) in Discussion