We study the mean curvature flow in 3-dimensional null hypersurfaces. In a spacetime a hypersurface is called null, if its induced metric is degenerate. The speed of the mean curvature flow of spacelike surfaces in a null hypersurface is the projection of the codimension-two mean curvature vector onto the null hypersurface. We impose fairly mild conditions on the null hypersurface. Then for an outer un-trapped initial surface, a condition which resembles the mean-convexity of a surface in Euclidean space, we prove that the mean curvature flow exists for all times and converges smoothly to a marginally outer trapped surface (MOTS). As an application we obtain the existence of a global foliation of the past of an outermost MOTS, provided the null hypersurface admits an un-trapped foliation asymptotically.
We define an explicit quasi-local mass functional which is nondecreasing along all foliations (satisfying a convexity assumption) of null cones. We use this new functional to prove the null Penrose conjecture under fairly generic conditions.
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In this paper, we survey recent progress on the Null Penrose Conjecture, including a proof of the conjecture for smooth null cones that are foliated by doubly convex spheres.
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