Rigidity of marginally outer trapped 2-spheres

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 $\tau\le0$ in a 4-dimensional spacetime $(\overline M,\overline g)$ when (under suitable assumptions) $M$ has a stable spherical MOTS $\Sigma$ 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 $\Sigma$, the metric $g$ splits as the product $([0,\varepsilon)\times\Sigma,dt^2+g_0)$, where $(\Sigma,g_0)$ is a round 2-sphere. In fact, we prove that the outer neighborhood $U$ can be isometrically embedded into the 4-dimensional Nariai spacetime $(\overline N,\overline h)$ as a spacelike hypersurface so that $g|_U$ is the induced metric from $\overline N$ and $K|_U$ is the second fundamental form of $U$ in $\overline N$. This completely classifies the local geometry of such initial data sets.