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

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.