We provide a direct proof of a noncollapsing estimate for compact hypersurfaces with positive mean curvature moving under the mean curvature flow: Precisely, if every point on the initial hypersurface admits an interior sphere with radius inversely proportional to the mean curvature at that point, then this remains true for all positive times in the interval of existence.
53C44; 58J35, 35K93We follow Sheng and Wang [4] in defining a notion of "noncollapsing" for embedded hypersurfaces as follows: Recall that a hypersurface M is called mean-convex if the mean curvature H of M is positive everywhere. It was proved in [4] that any compact mean-convex solution of the mean curvature flow is ı -noncollapsed for some ı > 0. Closely related statements are deduced by Brian White in [6]. In both of these works the result is derived only after a lengthy analysis of the properties of solutions of mean curvature flow. The purpose of this paper is to provide a self-contained proof of such a noncollapsing result using only the maximum principle.
A complete classification is given of curves in the plane which contract homothetically when evolved according to a power of their curvature. Applications are given to the limiting behaviour of the flows in various situations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.