# Singularity Profile in the Mean Curvature Flow - Mathematics > Differential Geometry

Abstract: In this paper we study the geometry of first time singularities of the meancurvature 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 ablow-up sequence converges locally smoothly along a subsequence to a smooth,convex blow-up solution. As a consequence we obtain a local Harnack inequalityfor the mean convex flow.

Author: Weimin Sheng, Xu-Jia Wang

Source: https://arxiv.org/