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 . Some parts are also inspired by the work of Perelman [32,33]. In a forthcoming paper , 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 ℝ3 by Brendle and Huisken  in September 2013 and in another paper by the authors in April 2014.
ASJC Scopus subject areas
- Applied Mathematics