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Collapsing and noncollapsing in convex ancient mean curvature flow

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arxiv 2106.06339 v2 pith:IH3IB5PF submitted 2021-06-11 math.DG

Collapsing and noncollapsing in convex ancient mean curvature flow

classification math.DG
keywords collapsingflowconvexcurvaturemeanancientmodelsnoncollapsing
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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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