AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces
studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time.
But as {t\rightarrow-\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\leq J\leq n-1}.
These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.
Ancient solutions for Andrews’ hypersurface flow