Paper · Sep 1, 2026

Quantitative avoidance for free boundary flows and applications

Yueheng Bao, Robert Haslhofer

math.DGWider flowsarXiv:2609.01026
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Abstract

In this article, we introduce a new distance function between hypersurfaces with free boundary. We show that our new quantity, which we call twisted Fermi distance, is monotone under mean curvature flow with free boundary. This overcomes the stumbling block that monotonicity of the usual distance function can fail for non-convex domains, and has several applications. Most importantly, we generalize the avoidance principle for free boundary Brakke flows, recently established by the first author for convex domains, to arbitrary domains. Using this, we then show that all results from our recent joint work, including the mean-convex neighborhood theorem and the uniqueness theorem for free boundary flows through cylindrical singularities, can be generalized to arbitrary domains without any convexity assumptions as well.

Comments: 11 pages

Mean curvature flowSingularitiesWeak & singular flowsStability & 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 Free boundary flow with surgery (Robert Haslhofer, 2023)2Haslhofer 2023 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)5Haslhofer 2023 On κ-solutions and canonical neighborhoods in 4d Ricci flow (Robert Haslhofer, 2023)8Haslhofer 2023 Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains (Theodora Bourni, Nathan Burns and Spencer Catron, 2024)7Bourni 2024 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 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)3Ghosh 2026 On the Gage-Hamilton-Grayson Theorem for free boundary curves (Yueheng Bao and Lorenzo Sarnataro, 2026)4Bao 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, Weak & singular flows, Singularities · Same authors Yueheng Bao, Robert Haslhofer · Shared terms “convex neighborhood”, “cylindrical singularities”, “boundary Brakke”

  2. Free boundary flow with surgery

    Robert HaslhoferWider flows PDF

    Topics Free boundary & capillary, Singularities · Same author Robert Haslhofer · Shared terms “mean convex”

  3. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows PDF

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

  4. On the Gage-Hamilton-Grayson Theorem for free boundary curves

    Yueheng Bao and Lorenzo SarnataroWider flows PDF

    Topics Free boundary & capillary, Singularities · Same author Yueheng Bao · Shared terms “mean convex”

  5. Free boundary minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows PDF

    Topics Free boundary & capillary · Same author Robert Haslhofer · Shared terms “mean convex”

  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 · Shared terms “convex domains”

  7. Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

    Theodora Bourni, Nathan Burns and Spencer CatronWider flows PDF

    Topics Free boundary & capillary · Shared terms “convex domains”

  8. On -solutions and canonical neighborhoods in 4d Ricci flow

    Robert Haslhofer PDF

    Topics Weak & singular flows, Singularities, Stability & uniqueness · Same author Robert Haslhofer · Shared terms “neighborhood theorem”, “convex neighborhood”, “cylindrical 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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 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.