Paper · Mar 14, 2025

How close is too close for singular mean curvature flows?

Joshua Daniels-Holgate, Or Hershkovits

math.DGWider flowsarXiv:2503.11522
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Abstract

Suppose , , are two mean curvature flows in encountering a multiplicity one compact singularity at time , in such a manner that for every , the Hausdorff distance between the two flows, , satisfies . We demonstrate that for every . This generalizes a result of Martin-Hagemayer and Sesum, who proved the case where is itself a self-similarly shrinking flow.

Mean curvature flowSingularities

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    Topics Singularities · Shared terms “compact singularity”, “multiplicity”

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  6. Non-uniqueness in Mean Curvature Flow: Non-canonical solutions via the parabolic Allen–Cahn

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

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  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.