Define BH boundary via cores and QLHs in general spacetimes

Prove that in general (non-spherically symmetric) spacetimes admitting trapped surfaces there exists a preferred core whose boundary is a preferred quasi-local horizon surface, thereby providing a definitive quasi-local definition of a dynamical black hole boundary; begin by extending the spherical results to axisymmetric spacetimes.

Background

In spherical symmetry, the boundary of the unique spherical core coincides with the unique spherical QLH, offering a clean quasi-local characterization of the black hole boundary. Beyond spherical symmetry, trapped regions and their boundaries exhibit nonlocal features analogous to EH teleology, complicating a canonical definition.

Establishing existence of a preferred core and corresponding QLH in axisymmetric and then general spacetimes would resolve foundational questions—what a dynamical black hole is and where its boundary lies—within a quasi-local geometric framework.