Paper · Mar 6, 2017

The moduli space of two-convex embedded tori

Reto Buzano, Robert Haslhofer, Or Hershkovits

math.DGWider flowsarXiv:1703.01758
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Abstract

In this short article we investigate the topology of the moduli space of two-convex embedded tori . We prove that for this moduli space is path-connected, and that for the connected components of the moduli space are in bijective correspondence with the knot classes associated to the embeddings. Our proof uses a variant of mean curvature flow with surgery developed in our earlier article (arXiv:1607.05604) where neck regions are deformed to tiny strings instead of being cut out completely, an approach which preserves the global topology, embeddedness, as well as two-convexity.

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Submitted
Mar 6, 2017
Subjects
math.DG, math.GT
Journal
Int. Math. Res. Not. IMRN 2019 (2), 392-406
DOI
10.1093/imrn/rnx125

Comments: 12 pages

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 spheres (Reto Buzano, Robert Haslhofer and Or Hershkovits, 2016)1Buzano 2016 Neck Detection for Two-Convex Hypersurfaces Embedded in Euclidean Space undergoing Brendle-Huisken G-Flow (Alexander Majchrowski, 2018)4Majchrowski 2018 Ancient low entropy flows, mean convex neighborhoods, and uniqueness (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2018)8Choi 2018 Ancient asymptotically cylindrical flows and applications (Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al., 2019)6Choi 2019 A note on the selfsimilarity of limit flows (Beomjun Choi, Robert Haslhofer and Or Hershkovits, 2019)5Choi 2019 A note on Alexandrov immersed mean curvature flow (Ben Lambert and Elena Mäder-Baumdicker, 2022)7Lambert 2022 Free boundary flow with surgery (Robert Haslhofer, 2023)3Haslhofer 2023 A gradient estimate for the linearized translator equation (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2025)2Choi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The moduli space of two-convex embedded spheres

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Surgery · Same authors Reto Buzano, Robert Haslhofer, Or Hershkovits · Shared terms “convex embedded”, “moduli space”, “path connected”

  2. A gradient estimate for the linearized translator equation

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Mean curvature flow · Same authors Robert Haslhofer, Or Hershkovits · Shared terms “moduli space”

  3. Free boundary flow with surgery

    Robert HaslhoferWider flows

    Topics Surgery · Same author Robert Haslhofer

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

    Alexander MajchrowskiWider flows

    Topics Surgery

  5. A note on the selfsimilarity of limit flows

    Beomjun Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Mean curvature flow · Same authors Robert Haslhofer, Or Hershkovits

  6. Ancient asymptotically cylindrical flows and applications

    Kyeongsu Choi, Robert Haslhofer, Or Hershkovits et al.Wider flows

    Topics Mean curvature flow · Same authors Robert Haslhofer, Or Hershkovits

  7. A note on Alexandrov immersed mean curvature flow

    Ben Lambert and Elena Mäder-BaumdickerWider flows

    Topics Surgery · Shared terms “convex embedded”

  8. Ancient low entropy flows, mean convex neighborhoods, and uniqueness

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics Mean curvature flow · Same authors Robert Haslhofer, Or Hershkovits

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.