Intrinsic rigidity for extremal Killing horizons with nontrivial bundle topology

Determine whether intrinsic rigidity holds for extremal Killing horizons whose bundle structure is nontrivial, and establish how the null convergence condition can be interpreted in terms of horizon data in that setting.

Background

The paper proves intrinsic rigidity for near-horizon geometries under a trivial bundle assumption: the relevant null hypersurfaces are required to be diffeomorphic to a trivial bundle whose fibers are generated by the null vector field, with compact and connected base. The outlook asks whether the rigidity result survives when this topological assumption is removed.

For horizons with nontrivial bundle structure, the constraint equation is reported to coincide with the trivial-bundle case, but the authors state that the interpretation of the null convergence condition in terms of horizon data is unknown. Extending the geometric argument therefore requires resolving this missing interpretation.

References

In this work, we considered only extremal Killing horizons with trivial topology. It is an interesting question whether intrinsic rigidity also holds for extremal Killing horizons that lack a trivial bundle structure. Non-extremal Killing horizons of non-trivial topology appear as Cauchy horizons in Taub-NUT spacetimes. The role of the extremal case is unclear; nonetheless, these objects are interesting in their own right. Axisymmetric electro-vacuum solutions in four spacetime dimensions are classified , and no non-axisymmetric solution is known. When the horizon has a non-trivial bundle structure, the constraint equation is exactly the same as in the trivial bundle case ; however, the interpretation of the null convergence condition in terms of horizon data is unknown.

— Geometric origin of intrinsic rigidity for extremal horizons  (2608.30759 - Kamiński et al., 31 Aug 2026) in Section 'Summary and outlook'