Paper · Sep 16, 2026

Morse resolution of mean curvature flows with cylindrical singularities

Richard H. Bamler, Felix Schulze, Lu Wang

math.DGWider flowsarXiv:2609.18687
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Abstract

We show that a compact mean curvature flow whose singularities have multiplicity-one cylindrical tangent flows admits a smooth Morse resolution. The resolution agrees with the spacetime track outside any prescribed neighborhood of its singular set and all its critical points have the expected index. No nondegeneracy or isolatedness assumption is required. In the mean-convex case the result follows by smoothing the global arrival function.

Comments: 6 pages; comments welcome

Mean curvature flowSingularities

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    Topics Singularities · Shared terms “cylindrical singularities”, “cylindrical tangent”, “tangent flow”

  2. Classification of ancient cylindrical mean curvature flows and the Mean Convex Neighborhood Conjecture

    Richard H. Bamler and Yi LaiWider flows PDF

    Topics Singularities · Same author Richard H. Bamler · Shared terms “cylindrical singularities”, “mean convex”, “neighborhood”

  3. Mean curvature flow with generic initial data II

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    Topics Singularities · Same author Felix Schulze · Shared terms “tangent flow”, “multiplicity”

  4. Generic mean curvature flows with cylindrical singularities I: the normal forms and nondegeneracy

    Ao Sun and Jinxin XueWider flows PDF

    Topics Singularities · Shared terms “cylindrical singularities”, “nondegeneracy”, “mean convex”

  5. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows PDF

    Topics Singularities · Same author Felix Schulze · Shared terms “resolution”

  6. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows PDF

    Topics Singularities · Same author Richard H. Bamler · Shared terms “multiplicity”

  7. Singularities of mean curvature flow with bounded mean curvature and Morse index

    Yongheng HanWider flows PDF

    Topics Singularities · Shared terms “Morse”, “multiplicity”

  8. Revisiting generic mean curvature flow in

    Otis Chodosh, Kyeongsu Choi, Christos Mantoulidis et al.Wider flows PDF

    Topics Singularities · Same author Felix Schulze · Shared terms “multiplicity”, “tangent flow”

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.