Paper · Apr 23, 2021

Approximation of mean curvature flow with generic singularities by smooth flows with surgery

Joshua Daniels-Holgate

math.DGWider flowsarXiv:2104.11647v3
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Abstract

We construct smooth mean curvature flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We conclude by improving the entropy bound on the low-entropy Schoenflies conjecture.

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Submitted
Apr 23, 2021
Last revised
Oct 14, 2022 (v3)
Subjects
math.DG, math.AP
Journal
Advances in Mathematics, Volume 410, Part A, 3 December 2022, 108715 2022,
DOI
10.1016/j.aim.2022.108715

Comments: 37 pages. Final version, to appear in Advances in Mathematics

Mean curvature flowSingularitiesSurgery

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 2016, 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 The moduli space of two-convex embedded tori (Reto Buzano, Robert Haslhofer and Or Hershkovits, 2017)7Buzano 2017 A finiteness theorem via the mean curvature flow with surgery (Alexander Mramor, 2017)6Mramor 2017 Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow (Alexander Majchrowski, 2018)5Majchrowski 2018 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)2Mramor 2018 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)4Sun 2025 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)8Hughes 2025 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)1Székelyhidi 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)3Székelyhidi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows

    Topics Surgery, Singularities · Shared terms “neck pinch”, “pinch singularity”, “spherical”

  2. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider flows

    Topics Surgery · Shared terms “low entropy”

  3. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows

    Topics Surgery, Singularities · Shared terms “Haslhofer”

  4. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider flows

    Topics Surgery, Singularities

  5. Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow

    Alexander MajchrowskiWider flows

    Topics Surgery

  6. A finiteness theorem via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Surgery

  7. The moduli space of two-convex embedded tori

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Surgery

  8. Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery

    John Hughes

    Topics Surgery, 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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, 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)

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).