On uniqueness and nonuniqueness of ancient ovals

In this paper, we prove that any nontrivial $$\mathrm{S}\mathrm{O}$(k )\times $\mathrm{S}\mathrm{O}$(n+1-k)$-symmetric ancient compact noncollapsed solution of the mean curvature flow agrees up to scaling and rigid motion with the $\mathrm{O}(k)\times \mathrm{O}(n+1-k)$-symmetric ancient ovals constructed by Hershkovits and the second author. This confirms a conjecture by Angenent-Daskalopoulos-Sesum. On the other hand, for every $k\ge 2$ we also construct a $(k-1)$-parameter family of uniformly $(k+1)$-convex ancient ovals that are only ${\mathbb{Z}}_2^k\times \mathrm{O}(n+1-k)$-symmetric. This gives counterexamples to a conjecture of Daskalopoulos.