Rigidity of marginally outer trapped 2-spheres

This is the authors' abstract. We don't add key points for this paper.

In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for a 3-dimensional initial data set $(M,g,K)$ with mean curvature τ≤0 in a 4-dimensional spacetime ( M, g) when (under suitable assumptions) M has a stable spherical MOTS $Σ$ which achieves an upper bound for the area. He proved that each leaf is a round 2-sphere with the same constant Gaussian curvature. Here, we improve his result by dropping the hypothesis on the mean curvature of M and showing that each leaf is a minimal surface. We show that in an outer neighborhood U of $Σ$, the metric g splits as the product ([0,ε)×Σ,dt²+g₀), where (Σ,g₀) is a round 2-sphere. In fact, we prove that the outer neighborhood U can be isometrically embedded into the 4-dimensional Nariai spacetime ( N, h) as a spacelike hypersurface so that g|U is the induced metric from N and K|U is the second fundamental form of U in N. This completely classifies the local geometry of such initial data sets.