Paper · Sep 17, 2026

A stable self-shrinking Möbius bundle in

Kahnrad Braxton, Tang-Kai Lee, Jonathan J. Zhu

math.DGarXiv:2609.19648
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Abstract

We prove the linear stability of an embedded mean curvature flow shrinker in with the topology of the Möbius bundle. This provides a non-flat, non-spherical stable shrinker in higher codimension, in sharp contrast with the classification of stable hypersurface shrinkers. Topologically, the Möbius shrinker models the reversal of a real blow-up, suggesting that stable higher-codimension singularities may encode non-trivial topological operations under mean curvature flow.

Comments: 23 pages, comments welcome

Mean curvature flowRicci solitonsSelf-shrinkers & translatorsSingularitiesStability & uniqueness

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Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Compactness and rigidity of self-shrinking surfaces (Tang-Kai Lee, 2021)1Lee 2021 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)7Lee 2023 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)8Chu 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)3Hirsch 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)2Lee 2024 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)5Lee 2025 Topology of gradient Ricci shrinkers via weighted L² cohomology (Fei He, 2026)6He 2026 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)4Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Compactness and rigidity of self-shrinking surfaces

    Tang-Kai LeeWider flows PDF

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Tang-Kai Lee · Shared terms “self shrinking”, “higher codimension”

  2. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao PDF

    Topics Self-shrinkers & translators, Singularities, Ricci solitons · Same author Tang-Kai Lee

  3. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows PDF

    Topics Self-shrinkers & translators, Singularities, Stability & uniqueness · Same author Jonathan J. Zhu

  4. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al. PDF

    Topics Singularities, Ricci solitons · Same author Tang-Kai Lee · Shared terms “embedded mean”

  5. Planarity and convexity for pinched ancient solutions of mean curvature flow

    Tang-Kai Lee, Keaton Naff and Jingze Zhu PDF

    Topics Ricci solitons · Same author Tang-Kai Lee · Shared terms “higher codimension”

  6. Topology of gradient Ricci shrinkers via weighted cohomology

    Fei He PDF

    Topics Self-shrinkers & translators, Ricci solitons

  7. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Singularities, Stability & uniqueness · Same author Tang-Kai Lee

  8. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Self-shrinkers & translators, Singularities · Shared terms “contrast”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  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.