On the extension to mean curvature flow in lower dimension

TL;DR

This paper proves that certain stable solutions to mean curvature flow in dimensions 2 to 6 with bounded mean curvature can be smoothly extended beyond finite singularity times.

Contribution

It establishes a new extension criterion for mean curvature flow solutions in lower dimensions based on stability and bounded mean curvature.

Findings

Stable solutions in dimensions 2 to 6 can be extended past finite singularities.

Bounded mean curvature is sufficient for smooth extension.

Extension results apply to embedded closed solutions with stability.

Abstract

In this paper, we prove that if $M_t\subset \mathbb{R}^{n+1}$, $2\leq n\leq 6$, is the $n$-dimensional closed embedded $\mathcal{F}-$stable solution to mean curvature flow with mean curvature of $M_t$ is uniformly bounded on $[0,T)$ for $T<\infty$, then the flow can be smoothly extended over time $T$.