Mean curvature flow of mean convex hypersurfaces
Abstract
In the last 15 years, White and Huisken-Sinestrari developed a far-reaching structure theory for the mean curvature flow of mean convex hypersurfaces. Their papers provide a package of estimates and structural results that yield a precise description of singularities and of high curvature regions in a mean convex flow. In the present paper, we give a new treatment of the theory of mean convex (and k-convex) flows. This includes: (1) an estimate for derivatives of curvatures, (2) a convexity estimate, (3) a cylindrical estimate, (4) a global convergence theorem, (5) a structure theorem for ancient solutions, and (6) a partial regularity theorem. Our new proofs are both more elementary and substantially shorter than the original arguments. Our estimates are local and universal. A key ingredient in our new approach is the new non- collapsing result of Andrews. Some parts are also inspired by the work of Perelman. In a forthcoming paper, we will give a new construction of mean curvature flow with surgery based on the theorems established in the present paper.
Keywords
Cite
@article{arxiv.1304.0926,
title = {Mean curvature flow of mean convex hypersurfaces},
author = {Robert Haslhofer and Bruce Kleiner},
journal= {arXiv preprint arXiv:1304.0926},
year = {2014}
}
Comments
Minor changes to facilitate applications to mean curvature flow with surgery