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

TL;DR

This paper proves that certain low-entropy hypersurfaces in four-dimensional space maintain connectivity under level set flow, and establishes a precise forward clearing out lemma for such flows.

Contribution

It demonstrates that low-entropy hypersurfaces in -dimensional space do not disconnect during level set flow and provides a sharp version of the forward clearing out lemma.

Findings

Level set flow of low-entropy hypersurfaces remains connected.

Established a sharp forward clearing out lemma for non-fattening flows.

Identified entropy threshold 1b0(2b2b7b2) for connectivity preservation.

Abstract

We show that if $\Sigma\subset \mathbb R^4$ is a closed, connected hypersurface with entropy $\lambda(\Sigma)\leq \lambda(\mathbb{S}^2\times \mathbb R)$, 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.