An application of Brakke's local regularity theorem

Mean curvature flow starting from plane-like manifolds I

Mean curvature flow starting from plane-like manifolds II

Mean curvature flow starting from plane-like manifolds III

Mean curvature flow starting from plane-like manifolds IV

"Null mean curvature" flow and marginally outer trapped surfaces
In this talk we discuss a new second order parabolic evolution equation for hypersurfaces in space-time initial data sets, that generalizes mean curvature flow (MCF). In particular, the `null mean curvature' - a space-time extrinsic curvature quantity - replaces the usual mean curvature in the evolution equation defining MCF. This flow is motivated by the study of black holes and mass/energy inequalities in general relativity. We present a theory of weak solutions using level-set methods and an appropriate variational principle, and outline a natural application of the flow as a parabolic approach to finding outermost marginally outer trapped surfaces (MOTS), which play the role of quasi-local black hole boundaries in general relativity. This is joint work with Kristen Moore.