Paper · Jun 13, 2023

Free boundary flow with surgery

Robert Haslhofer

math.DGWider flowsarXiv:2306.07714Geom. Topol. 30 (2026) 1-22
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Abstract

In this paper, we prove the existence of mean curvature flow with surgery for mean-convex surfaces with free boundary. To do so, we implement our recent new approach for constructing flows with surgery without a prior estimates in the free boundary setting. The flow either becomes extinct in finite time or for converges smoothly in the one or two sheeted sense to a finite collection of stable connected minimal surfaces with empty or free boundary (in particular, there are no surgeries for sufficiently large). Our free boundary flow with surgery will be applied in forthcoming work with Ketover, where we will address the existence problem for free boundary minimal disks in convex balls.

Comments: 20 pages. arXiv admin note: substantial text overlap with arXiv:2305.16267

Mean curvature flowSingularitiesSurgeryFree boundary & capillaryLong-time existence & convergenceMinimal & CMC surfaces

Its neighbourhood

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 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)7Chai 2022 Free boundary minimal disks in convex balls (Robert Haslhofer and Daniel Ketover, 2023)1Haslhofer 2023 Ancient solutions to free boundary mean curvature flow (Theodora Bourni and Giada Franz, 2026)4Bourni 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)8Székelyhidi 2026 Free boundary flow through cylindrical singularities (Yueheng Bao and Robert Haslhofer, 2026)2Bao 2026 Free boundary flows by powers of the Gauss curvature in the unit ball (Tianci Luo, Yong Wei and Rong Zhou, 2026)6Luo 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)3Bao 2026 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)5Ghosh 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 minimal disks in convex balls

    Robert Haslhofer and Daniel KetoverWider flows PDF

    Topics Minimal & CMC surfaces, Free boundary & capillary · Same author Robert Haslhofer · Shared terms “convex balls”, “minimal disk”, “mean convex”

  2. Free boundary flow through cylindrical singularities

    Yueheng Bao and Robert HaslhoferWider flows PDF

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

  3. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows PDF

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

  4. Ancient solutions to free boundary mean curvature flow

    Theodora Bourni and Giada FranzWider flows PDF

    Topics Minimal & CMC surfaces, Free boundary & capillary, Long-time existence & convergence · Shared terms “mean convex”

  5. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows PDF

    Topics Free boundary & capillary, Singularities

  6. Free boundary flows by powers of the Gauss curvature in the unit ball

    Tianci Luo, Yong Wei and Rong ZhouWider flows PDF

    Topics Free boundary & capillary, Long-time existence & convergence, Singularities · Shared terms “becomes extinct”, “converges smoothly”, “finite time”

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

    Xiaoxiang ChaiWider flows PDF

    Topics Free boundary & capillary, Singularities · Shared terms “finite time”

  8. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows PDF

    Topics Surgery, Singularities · Shared terms “mean convex”

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).