Stability and rigidity of axisymmetric marginally outer trapped two-spheres

TL;DR

This paper investigates the stability and rigidity of axisymmetric marginally outer trapped surfaces (MOTS) in initial data sets with symmetries, extending previous results to more specific geometric conditions and exploring their implications in rotating Nariai spacetimes.

Contribution

It refines existing rigidity results for MOTS by focusing on axisymmetric cases and introduces new conditions for stability and a foliation lemma for such surfaces.

Findings

Derived stability conditions for axisymmetric MOTS.

Established a foliation lemma for surfaces with constant outward null expansion.

Connected results to properties of rotating Nariai spacetimes.

Abstract

In [7], H. Bray, S. Brendle, and A. Neves studied rigidity properties of area-minimizing two-spheres in Riemannian three-manifolds with uniformly positive scalar curvature. In [13], these results were extended to marginally outer trapped surfaces (MOTS) in general initial data sets $(M^3,g,K)$ under a natural energy condition. In the present work, we refine the latter results to the setting of axisymmetric MOTS in initial data sets admitting a nontrivial Killing vector field. Conditions for the stability of such MOTS, as well as a new foliation lemma by axisymmetric surfaces of constant outward null expansion, are obtained. Finally, we discuss some aspects of the rotating Nariai spacetimes and their relation to these results.