The mean curvature at the first singular time of the mean curvature flow

Abstract

Consider a family of smooth immersions $F(\cdot ,t):M^n\to {\mathbb{R}}^{n+1}$ of closed hypersurfaces in ${\mathbb{R}}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t}=-H(p,t)\cdot \nu (p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singularities are of type I. In the case $n=2$, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds.

References in corpus (3)

• Extend Mean Curvature Flow with Finite Integral Curvature
• A Characterization of the Singular Time of the Mean Curvature Flow
• The Extension for Mean Curvature Flow with Finite Integral Curvature in Riemannian Manifolds

Cited by in corpus (13)

• On Type I Singularities in Ricci flow
• On the conditions to extend Ricci flow(II)
• Blow-up of the mean curvature at the first singular time of the mean curvature flow
• Remarks on the extension of the Ricci flow
• Remarks on curvature behavior at the first singular time of the Ricci flow
• The extension problem of the mean curvature flow (I)
• Mean curvature flow of surfaces in a hyperkähler $4$-manifold
• On the extension of axially symmetric volume flow and mean curvature flow
• On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature
• Extension of two-dimensional mean curvature flow with free boundary
• Some extensions of the mean curvature flow in Riemannian manifolds
• Self-shrinkers with bounded HA
• Blow up of subcritical quantities at the first singular time of the mean curvature flow