Communications in Analysis and Geometry

Volume 29 (2021)

Number 7

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

Pages: 1523 – 1543



Jacob Bernstein (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)

Shengwen Wang (Mathematics Institute, University of Warwick, Coventry, United Kingdom)


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.

The first author was partially supported by the NSF Grant DMS-1609340.

Received 19 February 2018

Accepted 3 April 2019

Published 17 May 2022