Ancient solutions for Andrews’ hypersurface flow

Abstract

Abstract We construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994. As time t → 0 - {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as t → - ∞ {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 × ℝ n - J {S^{J}\times\mathbb{R}^{n-J}} , 1 ≤ J ≤ n - 1 {1\leq J\leq n-1} . These results are the analog of the corresponding results in Ricci flow ( J = n - 1 {J=n-1} ) and mean curvature flow.