Communications in Analysis and Geometry

Volume 25 (2017)

Number 3

Almost graphical hypersurfaces become graphical under mean curvature flow

Pages: 589 – 623

DOI: https://dx.doi.org/10.4310/CAG.2017.v25.n3.a4

Author

Ananda Lahiri (Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam-Golm, Germany; and Fachbereich Mathematik und Informatik, Institut für Mathematik, Freie Universität Berlin, Germany)

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 in 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.

Full Text (PDF format)

Received 2 June 2015

Published 13 September 2017