The level set flow of a hypersurface in ${\mathbb{R}}^4$ of low entropy does not disconnect

Abstract

We show that if $\Sigma \subset {\mathbb{R}}^4$ is a closed, connected hypersurface with entropy $\lambda (\Sigma )\le \lambda ({\mathbb{S}}^2\times \mathbb{R})$, then the level set flow of $\Sigma$ never disconnects. We also obtain a sharp version of the forward clearing out lemma for non-fattening flows in ${\mathbb{R}}^4$ of low entropy.

arXiv commentStrengthened statement of Proposition 3.4 and fixed a gap in proof of Proposition 3.5