Abstract

Mean curvature flow (henceforth MCF) is one of the most extensively geometric flows that evolves hypersurfaces in Euclidean space by its mean curvature vector.

One of the central topics in MCF theory is the classification of ancient solutions, possibly with an extra geometric assumption. In this talk, we discuss the rigidity of ancient MCF that is asymptotic to a minimal quadratic cone in high dimensions.

We first show that if such flow lies on one side of the cone, then it has unique asymptotics. We then show that under an additional mean convexity assumption, such flow has to be a stationary solution given by one of the leaves of the Hardt-Simon foliation. This is the first rigidity / unique asymptotics result for ancient MCF asymptotic to a singular model shrinker.

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