*Accepted Paper*

**Inserted:** 13 nov 2020

**Last Updated:** 13 nov 2020

**Journal:** Communications in Analysis and Geometry

**Year:** 2018

**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.