Methods and Applications of Analysis

Volume 16 (2009)

Number 2

Singularity Profile in the Mean Curvature Flow

Pages: 139 – 156

DOI: http://dx.doi.org/10.4310/MAA.2009.v16.n2.a1

Authors

Weimin Sheng

Xu-Jia Wang

Abstract

In this paper we study the geometry of first time singularities of the mean curvature flow. By the curvature pinching estimate of Huisken and Sinestrari, we prove that a mean curvature flow of hypersurfaces in the Euclidean space $Bbb R^n+1$ with positive mean curvature is $kappa$-noncollapsing, and a blow-up sequence converges locally smoothly along a subsequence to a smooth, convex blow-up solution. As a consequence we obtain a local Harnack inequality for the mean convex flow.

Keywords

Mean curvature flow; singularity profile; $kappa$-noncollapsing

2010 Mathematics Subject Classification

35K55, 53C44

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