Collapsing and noncollapsing in convex ancient mean curvature flow

2021/06/11 by Theodora Bourni, Bourni, Theodora, Mat Langford +3 · 1 citation
Mathematics · #53E10 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2106.06339

openalex publication_date 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We provide several characterisations of collapsing and noncollapsing in convex ancient mean curvature flow, establishing in particular that collapsing occurs if and only if the flow is asymptotic to at least one Grim hyperplane. As a consequence, we rule out collapsing singularity models in (n-1)-convex mean curvature flow (even when the initial datum is only immersed). Explicit counterexamples show that (n-1)-convexity is optimal. We are also able to rule out collapsing singularity models for suitably pinched solutions of higher codimension.

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