Almost graphical hypersurfaces become graphical under mean curvature flow

Abstract

Consider a mean curvature ow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the ow is graphical inside the cylinder of half the radius. Here we prove a lower bound on this period depending on the Lipschitz-constant of the initial graphical representation. This is used to deal with a mean curvature ow that lies inside a slab and is initially graphical inside a cylinder except for a small set. We show that such a ow will become graphical inside the cylinder of half the radius. The proofs are mainly based on White’s regularity theorem.