Ancient solutions for Andrews' hypersurface flow.

We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time t → 0− the solutions collapse to a round point where 0 is the singular time. But as t → − ∞ the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions SJ × ℝn−J, 1 ≤ J ≤ n − 1. These results are the analog of the corresponding results in Ricci flow ( J = n − 1) and mean curvature flow. [ABSTRACT FROM AUTHOR]