Methods and Applications of Analysis
Volume 16 (2009)
Singularity Profile in the Mean Curvature Flow
Pages: 139 – 156
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