Singularity Profile in the Mean Curvature Flow

TL;DR

This paper investigates the geometry of initial singularities in mean curvature flow, proving noncollapsing and smooth convergence to convex solutions, and establishing a local Harnack inequality for mean convex flows.

Contribution

It demonstrates that flows with positive mean curvature are noncollapsing and that blow-up sequences converge smoothly to convex solutions, providing new geometric insights.

Findings

Flow with positive mean curvature is κ-noncollapsing.

Blow-up sequences converge to smooth, convex solutions.

Establishes a local Harnack inequality for mean convex flow.

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 $\R^{n+1}$ 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.