### 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 language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - 2016 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21650

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Mean Curvature Flow of Mean Convex Hypersurfaces

AU - Haslhofer, Robert

AU - Kleiner, Bruce

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84976297947&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976297947&partnerID=8YFLogxK

U2 - 10.1002/cpa.21650

DO - 10.1002/cpa.21650

M3 - Article

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -