Paper · Mar 27, 2018

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

Alexander Majchrowski

math.DGWider flowsarXiv:1803.09878
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Abstract

Recently Brendle-Huisken introduced a fully nonlinear flow . Their aim was to extend the surgery algorithm of Huisken-Sinestrari, into the Riemannian setting. The aim of this paper is to go through the details on how to perform neck detection for a closed, embedded hypersurface in undergoing this -flow. In order to do this we make some adjustments to Brendle and Huiskens gradient estimate, after we have done this we can go on to argue as in Huisken-Sinestrari's paper Mean curvature flow with surgeries of two-convex hyperusrfaces, in order to classify two-convex surfaces undergoing -flow.

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Submitted
Mar 27, 2018
Subjects
math.DG
Mean curvature flowSurgery

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)3Buzano 2017 A finiteness theorem via the mean curvature flow with surgery (Alexander Mramor, 2017)5Mramor 2017 On Neck Singularities for 2-Convex Mean Curvature Flow (Alexander Majchrowski, 2017)1Majchrowski 2017 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)6Mramor 2018 Approximation of mean curvature flow with generic singularities by smooth flows with surgery (Joshua Daniels-Holgate, 2021)8Daniels-Holg 2021 A note on Alexandrov immersed mean curvature flow (Ben Lambert and Elena Mäder-Baumdicker, 2022)4Lambert 2022 Nondegenerate neck pinches along the mean curvature flow (Gábor Székelyhidi, 2026)7Székelyhidi 2026 Mean convex flows with surgery (Gábor Székelyhidi, 2026)2Székelyhidi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On Neck Singularities for 2-Convex Mean Curvature Flow

    Alexander MajchrowskiWider flows

    Topics Surgery · Same author Alexander Majchrowski · Shared terms “neck detection”, “convex hypersurfaces”, “Sinestrari”

  2. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows

    Topics Surgery · Shared terms “Brendle Huisken”, “Huisken Sinestrari”, “convex hypersurfaces”

  3. The moduli space of two-convex embedded tori

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Surgery

  4. A note on Alexandrov immersed mean curvature flow

    Ben Lambert and Elena Mäder-BaumdickerWider flows

    Topics Surgery · Shared terms “Brendle”, “Huisken”, “gradient estimates”

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

    Alexander MramorWider flows

    Topics Surgery · Shared terms “convex hypersurfaces”

  6. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider flows

    Topics Surgery · Shared terms “convex hypersurfaces”

  7. Nondegenerate neck pinches along the mean curvature flow

    Gábor SzékelyhidiWider flows

    Topics Surgery

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

    Joshua Daniels-HolgateWider flows

    Topics Surgery

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) 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.