Classification of bubble-sheet ovals in $\mathbb{R}^{4}$

TL;DR

This paper classifies bubble-sheet ovals in four-dimensional space under mean curvature flow, identifying all possible symmetric ancient solutions up to symmetry and scaling, marking a significant step in understanding complex geometric flows.

Contribution

It provides the first classification of non-cohomogeneity-one, non-selfsimilar ancient solutions for mean curvature flow in four dimensions.

Findings

Classifies all bubble-sheet ovals in $\

up to symmetry and scaling.

Identifies two main families of ancient ovals with specific symmetries.

Abstract

In this paper, we prove that any bubble-sheet oval for the mean curvature flow in $\mathbb{R}^4$, up to scaling and rigid motion, either is the $\textrm{O}(2)\times \textrm{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of $\mathbb{Z}_2^2\times \textrm{O}(2)$-symmetric ancient ovals constructed by the third and fourth author. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.