Methods and Applications of Analysis

Singularity Profile in the Mean Curvature Flow

Weimin Sheng and Xu-Jia Wang

Full-text: Open access


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.

Article information

Methods Appl. Anal., Volume 16, Number 2 (2009), 139-156.

First available in Project Euclid: 2 November 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C44: Geometric evolution equations (mean curvature flow, Ricci flow, etc.) 35K55: Nonlinear parabolic equations

Mean curvature flow singularity profile $kappa$-noncollapsing


Sheng, Weimin; Wang, Xu-Jia. Singularity Profile in the Mean Curvature Flow. Methods Appl. Anal. 16 (2009), no. 2, 139--156.

Export citation