Mean curvature flow of mean convex hypersurfaces

TL;DR

This paper offers a simplified, unified approach to the theory of mean convex mean curvature flow, providing new estimates, structure theorems, and regularity results that enhance understanding of singularities and high curvature regions.

Contribution

It introduces more elementary, shorter proofs and new local, universal estimates for mean convex flows, building on Andrews' non-collapsing result and inspired by Perelman's work.

Findings

Derivative curvature estimates established

Convexity and cylindrical estimates proven

Global convergence and partial regularity theorems obtained

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.