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Closed mean curvature flows with asymptotically conical singularities

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arxiv 2405.15577 v3 pith:QHV6OASD submitted 2024-05-24 math.DG math.AP

Tang-Kai Lee , Xinrui Zhao This is my paper · ORCID
classification math.DGmath.AP
keywords closedasymptoticallyconicalcurvatureembeddedflowflowsmean
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In this paper, we prove that for any asymptotically conical self-shrinker, there exists an embedded closed hypersurface such that the mean curvature flow starting from it develops a singularity modeled on the given shrinker. The main technique is the Wa\.zewski box argument, used by Stolarski in the proof of the corresponding theorem in the Ricci flow case. As a corollary, our construction, combined with the works of Angenent--Ilmanen--Vel\'azquez and Chodosh--Daniels-Holgate--Schulze, implies the existence of fattening level set flows starting from smooth embedded closed hypersurfaces. These provide examples related to a question asked by Evans--Spruck.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Self-expanders of positive genus

    math.DG 2025-07 reject novelty 8.0 of 10

    For a broad class of cones in R^3, complete connected self-expanders of any positive genus exist, providing mean curvature flows in which genus drops but does not reach zero.

  2. The Willmore Energy Landscape of Spheres and Avoidable Singularities of the Willmore Flow

    math.DG 2025-06 conditional novelty 8.0 of 10

    Every immersed 2-sphere with Willmore energy at most 12π admits an energy-nonincreasing regular homotopy to a round sphere or to a surface in the explicit family J, giving exactly four regular homotopy classes below 12π.

  3. An Intersection Principle for Mean Curvature Flow

    math.DG 2025-05 conditional novelty 8.0 of 10

    For two mean curvature flows in any dimension, the Hausdorff dimension of their intersection is non-increasing over time, and for level set flows with finitely many singularities, non-fattening is equivalent to this i...

  4. Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$

    math.DG 2025-07 conditional novelty 7.0 of 10

    For every half-integer J and sufficiently large integer m, the authors construct an embedded self-shrinker with 2J+1 ends and genus 2J(m-1) by gluing stacked planes with m catenoidal bridges between adjacent levels.

  5. Passing through nondegenerate singularities in mean curvature flows

    math.DG 2025-01 conditional novelty 7.0 of 10

    Mean curvature flows through nondegenerate cylindrical singularities undergo an isolated, graphical surgery event whose topology change equals an (n-k)-surgery, matching Morse level sets.

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