Construction of High Codimension Ancient Mean Curvature Flows

Abstract

We construct a class of compact ancient solutions to the mean curvature flow in Euclidean space with high codimension. In particular, we construct higher codimensional ancient curve shortening flows. Moreover, we characterize the asymptotic behavior of these solutions. Add on remark: the construction in this paper has been discovered by Altschuler D.-Altschuler S.-Angenent-Wu in [AAAW13].

10 pages, comments are welcomed! After the first version of this preprint was uploaded to arxiv, we discovered that the construction in this paper has been discovered by Altschuler D.-Altschuler S.-Angenent-Wu in [AAAW13]. Since the motivation of our preprint is different from [AAAW13], we still hope the readers could find some innovating ideas from this preprint