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

TL;DR

This paper classifies all noncollapsed singularity models for mean curvature flow of 3D hypersurfaces in four-dimensional space, identifying specific known translating solutions and a family of oval bowls.

Contribution

It provides a complete classification of noncollapsed translating hypersurfaces in , extending understanding of singularity models in mean curvature flow.

Findings

Every noncollapsed translating hypersurface in  is either  d-bowl, a 3d round bowl, or a 3d oval bowl from Hoffman-Ilmanen-Martin-White.

The classification covers all noncollapsed singularity models in .

The results unify known solutions and identify a new family of solutions.

Abstract

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$2d-bowl, or a 3d round bowl, or belongs to the one-parameter family of 3d oval bowls constructed by Hoffman-Ilmanen-Martin-White.