Uniqueness of axisymmetric DHSs

Establish sharper restrictions on dynamical horizon segments admissible in axisymmetric spacetimes and determine whether an axisymmetric spacetime that admits a DHS necessarily admits a unique axisymmetric DHS representing a given dynamical phase.

Background

The interplay between Killing symmetries and DHSs implies strong geometric constraints: generically, rotational Killing fields are tangential to marginally trapped surfaces (MTSs) on DHSs, with exceptions leading to stability issues and special null 3-flats. These results suggest DHSs in axisymmetry may be highly restricted.

A rigorous uniqueness statement for axisymmetric DHSs would align mathematical control with numerical findings and sharpen the quasi-local depiction of black hole boundaries in symmetric settings.