Singularity Profile in the Mean Curvature Flow

Details Singularity Profile in the Mean Curvature Flow Singularity Profile in the Mean Curvature Flow

2009-02-13

arxiv.org/abs/0902.2261v2

Mathematics

Differential Geometry

Scientific paper

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 $\R^{n+1}$ 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.

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