Paper · Aug 7, 2025

Cylindrical tangent flows in mean curvature flow

Sourav Ghosh

math.DGWider flowsarXiv:2508.05517v3
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Abstract

The only non-compact linearly stable singularity models for mean curvature flow are cylindrical, as shown by Colding-Minicozzi. The uniqueness of blowups at singularities modeled on cylinders was established by Colding-Minicozzi. They also proved rigidity results for cylindrical singularities in their earlier work. In this paper, we develop a different approach inspired by Székelyhidi to study cylindrical singularities and prove uniqueness and rigidity results.

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Mean curvature flowSingularitiesStability & uniquenessBoundaries & non-compact

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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 On the Stability of Cylindrical Singularities of the Mean Curvature Flow (Jingxuan Zhang, 2021)2Zhang 2021 A nonexistence result for rotating mean curvature flows in ℝ⁴ (Wenkui Du and Robert Haslhofer, 2022)8Du 2022 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)3Lee 2023 Rigidity Properties of Colding-Minicozzi Entropies (Jacob Bernstein, 2023)6Bernstein 2023 Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow (Hanbing Fang and Yu Li, 2025)4Fang 2025 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)1Ghosh 2026 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)5Sun 2026 Morse resolution of mean curvature flows with cylindrical singularities (Richard H. Bamler, Felix Schulze and Lu Wang, 2026)7Bamler 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows PDF

    Topics Singularities, Stability & uniqueness · Same author Sourav Ghosh · Shared terms “tangent flow”, “cylindrical singularities”, “singularities modeled”

  2. On the Stability of Cylindrical Singularities of the Mean Curvature Flow

    Jingxuan ZhangWider flows PDF

    Topics Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”, “Minicozzi”, “tangent flow”

  3. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Boundaries & non-compact, Singularities, Stability & uniqueness · Shared terms “tangent flow”

  4. Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

    Hanbing Fang and Yu Li PDF

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

  5. Mean curvature flows with cylindrical singularities II: stability and genericity

    Ao Sun and Jinxin XueWider flows PDF

    Topics Singularities, Stability & uniqueness · Shared terms “cylindrical singularities”

  6. Rigidity Properties of Colding-Minicozzi Entropies

    Jacob BernsteinWider flows PDF

    Topics Stability & uniqueness · Shared terms “Colding Minicozzi”

  7. Morse resolution of mean curvature flows with cylindrical singularities

    Richard H. Bamler, Felix Schulze and Lu WangWider flows PDF

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

  8. A nonexistence result for rotating mean curvature flows in

    Wenkui Du and Robert HaslhoferWider flows PDF

    Topics Singularities, Stability & uniqueness · Shared terms “Colding Minicozzi”, “tangent flow”, “singularity models”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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.