Singularity of mean curvature flow with bounded mean curvature and Morse index

TL;DR

This paper investigates the behavior of singularities in mean curvature flow with bounded mean curvature and Morse index, revealing that either the mean curvature or Morse index must become unbounded at the first singularity in certain dimensions.

Contribution

It establishes a new result linking bounded mean curvature and Morse index to the inevitability of blow-up at the first singularity in dimensions 3 to 6.

Findings

Either mean curvature or Morse index blows up at the first singularity.

The result applies to closed smooth embedded flows in Euclidean space.

Provides insight into the nature of singularities in mean curvature flow.

Abstract

We study the multiplicity of the singularity of mean curvature flow with bounded mean curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed smooth embedded mean curvature flow in $\mathbb{R}^{n+1}$.