Blow up of subcritical quantities at the first singular time of the mean curvature flow

TL;DR

This paper proves that certain subcritical geometric quantities involving the second fundamental form blow up at the first singular time of the mean curvature flow, improving previous results by incorporating a logarithmic factor.

Contribution

It introduces a log improvement to the blow-up analysis of subcritical quantities at the first singular time in mean curvature flow.

Findings

Subcritical quantities involving the second fundamental form blow up at singularity.

Logarithmic factors refine previous blow-up results.

Enhances understanding of singularity formation in 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 show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example $\int_{0}^{t} \int_{M_{s}} \frac{\abs{A}^{n + 2}}{log (2 + \abs{A})} d\mu ds,$ blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.