Ancient solutions for Andrews' hypersurface flow

Abstract

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time $t\to 0^{-}$ the solutions collapse to a round point where $0$ is the singular time. But as $t\to -\infty$ the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger-Gromov limits are round cylinder solutions $S^J\times {\mathbb{R}}^{n-J}$, $1\le J\le n-1$. These results are the analog of the corresponding results in Ricci flow ($J=n-1$) and mean curvature flow.