Mean Curvature Flow of Mean Convex Hypersurfaces

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 noncollapsing result of Andrews [2]. Some parts are also inspired by the work of Perelman [32,33]. In a forthcoming paper [17], we will give a new construction of mean curvature flow with surgery based on the methods established in the present paper. Note added in May 2015 . Since the first version of this paper was posted on arxiv in April 2013, the estimates have been used to construct mean convex flow with surgery in ℝ 3 by Brendle and Huisken [5] in September 2013 and in another paper by the authors in April 2014.© 2016 Wiley Periodicals, Inc.