Ancient Solutions of Geometric Flows with Curvature Pinching

Abstract

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find pinching conditions on the second fundamental form that characterise the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension greater than one, and for some nonlinear curvature flows of hypersurfaces.

Ancient Solutions of Codimension Two Surfaces With Curvature Pinching - Retracted

“…Recently, Risa and Sinestrari [22] derived rigidity results for ancient solutions of more general curvature flows, by considering either mean curvature flow in higher codimension, or the hypersurface case with more general speeds than mean curvature. They consider various kinds of flows and their results are in a similar spirit to Theorem 1.1, showing that a suitable uniform pinching condition characterises the shrinking sphere among convex ancient solutions.…”

“…We consider ancient solutions on codimension two surfaces with some different pinching conditions from [22]. Our pinching conditions are inspired by [7] and our result is given in the following theorem.…”

Pinched ancient solutions to the high codimension mean curvature flow

“…There is however an immersion of the Veronese surface into R 5 which shrinks homothetically under the mean curvature flow and satisfies |h| 2 = 5 6 |H | 2 , so one cannot hope to do better than c 2 = 5 6 . We note that in [23], a similar theorem was proven assuming the extra condition that the second fundamental form is uniformly bounded.…”

Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the three-sphere

“…Concerning the mean curvature flow, the study of existence, regularity and classification for certain flows in manifolds has been an active topic in geometric analysis. Huisken and Sinestrari [22] showed that closed mean convex ancient flows in the sphere S n are shrinking spherical caps, provided it satisfies a curvature pinching condition (see [39] and [28], and the references therein, for generalizations to higher codimension and to other space forms). In [4], P. Bryan, M.N.…”