Cambridge Journal of Mathematics

Volume 11 (2023)

Number 3

Classification of noncollapsed translators in $\mathbb{R}^4$

Pages: 563 – 698



Kyeongsu Choi (School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea)

Robert Haslhofer (Department of Mathematics, University of Toronto, Ontario, Canada)

or Hershkovits (Institute of Mathematics, Hebrew University, Jerusalem, Israel)


In this paper, we classify all noncollapsed singularity models for the mean curvature flow of $3$-dimensional hypersurfaces in $\mathbb{R}^4$ or more generally in $4$-manifolds. Specifically, we prove that every noncollapsed translating hypersurface in $\mathbb{R}^4$ is either $\mathbb{R} \times 2\textrm{d}$-bowl, or a $3\textrm{d}$ round bowl, or belongs to the one-parameter family of $3\textrm{d}$ oval bowls constructed by Hoffman–Ilmanen–Martin–White.

KC has been supported by KIAS Individual Grant MG078901 and a POSCO Science Fellowship. RH has been supported by an NSERC Discovery Grant and a Sloan Research Fellowship. OH has been supported by the Koret Foundation early career award and ISF grant 437/20.

Received 6 July 2021

Published 8 August 2023