Almost graphical hypersurfaces become graphical under mean curvature flow

TL;DR

This paper establishes a lower bound on the time a hypersurface under mean curvature flow remains graphical inside a cylinder, and shows that flows initially nearly graphical become fully graphical over time.

Contribution

It provides a quantitative bound on the persistence of graphicality under mean curvature flow and extends results to flows nearly graphical inside a slab.

Findings

Flow remains graphical for a time depending on initial Lipschitz constant.

Nearly graphical flows inside a slab become fully graphical inside a smaller cylinder.

Uses White's regularity theorem for proofs.

Abstract

Consider a mean curvature flow of hypersurfaces in Euclidean space, that is initially graphical inside a cylinder. There exists a period of time during which the flow 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 flow that lies inside a slab and is initially graphical inside a cylinder except for a small set. We show that such a flow will become graphical inside the cylinder of half the radius. The proofs are mainly based on White's regularity theorem.