Rigidity of marginally outer trapped $2$-spheres

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped $2$-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4 \pi / c$, where $c \gt 0$ is a lower bound on a natural energy-momentum term.We then consider the rigidity that results for stable, or weakly outermost, marginally outer trapped $2$-spheres that achieve this upper bound on the area. In particular, we prove a splitting result for $3$-dimensional initial data sets analogous to a result of Bray, Brendle and Neves [Rigidity of area-minimizing two-spheres in three-manifolds, Comm. Anal. Geom. 18 (2010), no. 4, 821–830] concerning area minimizing $2$-spheres in Riemannian $3$-manifolds with positive scalar curvature. We further show that these initial data sets locally embed as spacelike hypersurfaces into the Nariai spacetime. Connections to the Vaidya spacetime and dynamical horizons are also discussed.

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Received 18 June 2015

Published 31 January 2018