A note on the rigidity of marginally outer trapped 2-spheres

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4\pi \mathrm{/}c$, where $c>0$ is a lower bound on a natural energy momentum term. In this note we consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. The “canonical” dynamical horizon in Vaidya spacetime and certain spacelike hypersurfaces in Nariai spacetime provide illustrations of the main results. These results may be viewed as spacetime analogues of the rigidity results of Bray, Brendle and Neves [10] concerning area minimizing 2-spheres in Riemannian 3-manifolds with scalar curvature having positive lower bound.