Methods and Applications of Analysis

Volume 16 (2009)

Number 2

Singularity Profile in the Mean Curvature Flow

Pages: 139 – 156



Weimin Sheng

Xu-Jia Wang


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.


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

2010 Mathematics Subject Classification

35K55, 53C44

Published 1 January 2009