Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$

Abstract

Motivated by the limiting behavior of an explicit class of compact ancient curve shortening flows, we prove codimension bounds for ancient mean curvature flows by their tangent flow at $-\infty$, generalizing a theorem for cylinders in [CM19b]. In the case of the $m$-covered circle, we apply this bound to prove a strong rigidity theorem. Furthermore, we extend this paradigm by showing that under the assumption of sufficiently rapid convergence, a compact ancient mean curvature flow is identical to its tangent flow at $-\infty$.

17 pages. Modify the paper according to the referees' suggestion. In particular we remove Theorem 1.3 in the previous version, where there is a gap in the proof. Accepted by Communications in Contemporary Mathematics. arXiv admin note: text overlap with arXiv:1908.02688