Paper · May 9, 2017

A finiteness theorem via the mean curvature flow with surgery

Alexander Mramor

math.DGWider flowsarXiv:1705.03219v2
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Abstract

In this article, we use the recently developed mean curvature flow with surgery for 2 convex hypersurfaces to prove several isotopy existence and finally extrinsic finiteness results (in the spirit of Cheeger's compactness theorem) for the space of 2 convex hypersurfaces in .

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Submitted
May 9, 2017
Last revised
Aug 22, 2017 (v2)
Subjects
math.DG, math.GT

Comments: 30 pages, but many figures. Comments welcome!

Mean curvature flowSurgeryLimit spaces & RCD

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)6Buzano 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)4Majchrowski 2018 Low Entropy and the Mean Curvature Flow with Surgery (Alexander Mramor and Shengwen Wang, 2018)1Mramor 2018 Nonconvex Surfaces which Flow to Round Points (Alexander Mramor and Alec Payne, 2019)5Mramor 2019 Approximation of mean curvature flow with generic singularities by smooth flows with surgery (Joshua Daniels-Holgate, 2021)7Daniels-Holg 2021 Passing through nondegenerate singularities in mean curvature flows (Ao Sun, Zhihan Wang and Jinxin Xue, 2025)8Sun 2025 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. Low Entropy and the Mean Curvature Flow with Surgery

    Alexander Mramor and Shengwen WangWider 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. Mean convex flows with surgery

    Gábor SzékelyhidiWider flows

    Topics Surgery · Shared terms “convex hypersurfaces”

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

    Alexander MajchrowskiWider flows

    Topics Surgery · Shared terms “convex hypersurfaces”

  5. Nonconvex Surfaces which Flow to Round Points

    Alexander Mramor and Alec PayneWider flows

    Topics Limit spaces & RCD · Same author Alexander Mramor · Shared terms “convex hypersurfaces”

  6. The moduli space of two-convex embedded tori

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Surgery · Shared terms “developed”

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

    Joshua Daniels-HolgateWider flows

    Topics Surgery

  8. Passing through nondegenerate singularities in mean curvature flows

    Ao Sun, Zhihan Wang and Jinxin XueWider 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 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.