Mean Curvature Flow of Mean Convex Hypersurfaces

Robert Haslhofer, Bruce Kleiner

Research output: Contribution to journalArticle

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 awnd 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.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - 2016

Fingerprint

Mean Curvature Flow
Surgery
Hypersurface
Derivatives
Estimate
Curvature
Partial Regularity
Structure Theorem
Global Convergence
Convergence Theorem
Convexity
Singularity
Derivative
Theorem

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Mean Curvature Flow of Mean Convex Hypersurfaces. / Haslhofer, Robert; Kleiner, Bruce.

In: Communications on Pure and Applied Mathematics, 2016.

Research output: Contribution to journalArticle

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