Ancient mean curvature flow asymptotic to a minimal quadratic cone

Abstract

In this paper, we prove that a smooth, properly embedded ancient mean curvature flow that is asymptotic to an $O(n)\times O(m)$ symmetric minimal quadratic cone for $n+m\ge 10$, and lies on one side of the cone has to have unique asymptotics in the parabolic region. When additionally assuming mean convexity, we upgrade unique asymptotics to full uniqueness, and show that such flow has to be a stationary flow given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity/unique asymptotics theory for ancient mean curvature flow with a $\text{singular minimal cone as the asymptotic model}$.

63 pages, V4 : Journal submission version, fixed minor typos, and extended the main results to more general minimal quadratic cones using same ideas, comments welcome