Paper · Apr 11, 2018

Low Entropy and the Mean Curvature Flow with Surgery

Alexander Mramor, Shengwen Wang

math.DGWider flowsarXiv:1804.04115v5
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Abstract

In this article, we extend the mean curvature flow with surgery to mean convex hypersurfaces with entropy less than . In particular, 2-convexity is not assumed. Next we show the surgery flow with just the initial convexity assumption is possible and as an application we use the surgery flow to show that smooth -dimensional closed self shrinkers with entropy less than are isotopic to the round -sphere.

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Submitted
Apr 11, 2018
Last revised
Nov 26, 2020 (v5)
Subjects
math.DG, math.GT

Comments: Revised after referee's report. A number of small errors fixed and minor parts of the argument were altered

Mean curvature flowSelf-shrinkers & translatorsSurgery

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 A finiteness theorem via the mean curvature flow with surgery (Alexander Mramor, 2017)1Mramor 2017 Regularity and stability results for the level set flow via the mean curvature flow with surgery (Alexander Mramor, 2017)2Mramor 2017 Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow (Alexander Majchrowski, 2018)6Majchrowski 2018 An unknottedness result for self shrinkers with multiple ends (Alexander Mramor, 2020)8Mramor 2020 Approximation of mean curvature flow with generic singularities by smooth flows with surgery (Joshua Daniels-Holgate, 2021)4Daniels-Holg 2021 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)7Hughes 2025 Mean convex flows with surgery (Gábor Székelyhidi, 2026)5Székelyhidi 2026 On knotted self expanders of the mean curvature flow in R³ (Alexander Mramor and Shengwen Wang, 2026)3Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A finiteness theorem via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Surgery · Same author Alexander Mramor · Shared terms “convex hypersurfaces”

  2. Regularity and stability results for the level set flow via the mean curvature flow with surgery

    Alexander MramorWider flows

    Topics Surgery · Same author Alexander Mramor

  3. On knotted self expanders of the mean curvature flow in

    Alexander Mramor and Shengwen Wang

    Topics Self-shrinkers & translators · Same authors Alexander Mramor, Shengwen Wang · Shared terms “isotopic”

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

    Joshua Daniels-HolgateWider flows

    Topics Surgery · Shared terms “low entropy”

  5. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows

    Topics Surgery · Shared terms “mean convex”, “convex hypersurfaces”

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

    Alexander MajchrowskiWider flows

    Topics Surgery · Shared terms “convex hypersurfaces”

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

    John Hughes

    Topics Surgery

  8. An unknottedness result for self shrinkers with multiple ends

    Alexander MramorWider flows

    Topics Self-shrinkers & translators · Same author Alexander Mramor

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) 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) 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. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

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

  3. 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.