Collapsing and noncollapsing in convex ancient mean curvature flow

TL;DR

This paper characterizes collapsing and noncollapsing behaviors in convex ancient mean curvature flows, linking collapsing to asymptotic Grim hyperplanes and ruling out certain collapsing singularity models.

Contribution

It provides new characterizations of collapsing in convex ancient flows and establishes conditions under which collapsing singularities cannot occur.

Findings

Collapsing occurs iff the flow is asymptotic to a Grim hyperplane.

Collapse is ruled out in (n-1)-convex flows, even with immersed initial data.

Counterexamples show (n-1)-convexity is optimal for these results.

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.