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

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 $\text{O}(2)\times \text{O}(2)$-symmetric ancient oval constructed by Hershkovits and the fourth author, or belongs to the one-parameter family of ${\mathbb{Z}}_2^2\times \text{O}(2)$-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.

88 pages. In v2, we added details to the proof of Corollary 1.5. In v3, minor corrections are made following referee's suggestion. The paper will appear in Geom. Topol