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

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