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

TL;DR

This paper establishes codimension bounds and rigidity results for ancient mean curvature flows based on their tangent flow at negative infinity, extending previous theorems and providing conditions for flow uniqueness.

Contribution

It generalizes codimension bounds for ancient flows and proves a strong rigidity theorem for the m-covered circle, also showing flow uniqueness under rapid convergence.

Findings

Codimension bounds for ancient mean curvature flows are established.

A strong rigidity theorem is proved for the m-covered circle.

Flow uniqueness is shown under rapid convergence assumptions.

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 $-\infty$.