Rigidity of marginally outer trapped 2-spheres

Abstract

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 $\tau \le 0$ 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 $([0,\epsilon )\times \Sigma ,d{t}^2+g_0)$, 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.

arXiv commentThis paper has been withdrawn; it is superseded by the paper: arXiv:1506.00611, with G.J. Galloway, which combines the results of this paper with those in arXiv:1503.05540

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Cited by in corpus (3)

• Rigidity of outermost MOTS - the initial data version
• Area Estimates and Rigidity of Non-compact $H$-Surfaces in 3-Manifolds
• Upper Bound for Diameter of Cosmological Black Holes and Nonexistence of Black Strings