Paper · Sep 2, 2026

Uniqueness of tangent flows in free boundary flow

Sourav Ghosh

math.DGWider flowsarXiv:2609.03185
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Abstract

We prove uniqueness of tangent flows for mean curvature flow with free boundary at singularities modeled on half-cylindrical self-shrinkers. More precisely, if a tangent flow is of the form or , then it is unique. This provides the free-boundary analogue of the uniqueness theory for cylindrical singularities.

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessFree boundary & capillary

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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 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)8Chai 2022 Free boundary flow with surgery (Robert Haslhofer, 2023)4Haslhofer 2023 Cylindrical tangent flows in mean curvature flow (Sourav Ghosh, 2025)3Ghosh 2025 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)7Nguyen 2026 Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow (Theodora Bourni, Nathan Burns and Mat Langford, 2026)6Bourni 2026 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)1Bao 2026 Singularity formation and self-shrinkers for the parabolic Bernoulli problem (Hunter Liu, Sebastian Munoz and Zihui Zhao, 2026)5Liu 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)2Bao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows PDF

    Topics Free boundary & capillary, Singularities · Shared terms “half cylindrical”, “cylindrical singularities”

  2. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows PDF

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

  3. Cylindrical tangent flows in mean curvature flow

    Sourav GhoshWider flows PDF

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

  4. Free boundary flow with surgery

    Robert HaslhoferWider flows PDF

    Topics Free boundary & capillary, Singularities

  5. Singularity formation and self-shrinkers for the parabolic Bernoulli problem

    Hunter Liu, Sebastian Munoz and Zihui ZhaoWider flows PDF

    Topics Free boundary & capillary, Self-shrinkers & translators, Singularities · Shared terms “tangent flow”

  6. Stability of the Shrinking Semi-Circle Under the Free Boundary Curve Shortening Flow

    Theodora Bourni, Nathan Burns and Mat LangfordWider flows PDF

    Topics Free boundary & capillary, Singularities, Stability & uniqueness

  7. High Codimension Curve Shortening Flow with Free Boundary

    Huy The Nguyen and Artemis A. VogiatziWider flows PDF

    Topics Free boundary & capillary, Self-shrinkers & translators, Singularities

  8. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows PDF

    Topics Free boundary & capillary, Singularities

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) 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) 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)

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.