Methods and Applications of Analysis

Singularity Profile in the Mean Curvature Flow

Weimin Sheng and Xu-Jia Wang

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 2 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.maa/1257170933

Mathematical Reviews number (MathSciNet)
MR2563745

Zentralblatt MATH identifier
1184.53071

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

Keywords
Mean curvature flow singularity profile $kappa$-noncollapsing

Citation

Sheng, Weimin; Wang, Xu-Jia. Singularity Profile in the Mean Curvature Flow. Methods Appl. Anal. 16 (2009), no. 2, 139--156. https://projecteuclid.org/euclid.maa/1257170933


Export citation