Paper · Jul 19, 2016

The moduli space of two-convex embedded spheres

Reto Buzano, Robert Haslhofer, Or Hershkovits

math.DGWider flowsarXiv:1607.05604v2
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Abstract

We prove that the moduli space of 2-convex embedded n-spheres in R^n+1 is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.

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Submitted
Jul 19, 2016
Last revised
Nov 28, 2017 (v2)
Subjects
math.DG, math.GT
Journal
J. Differential Geom. 118:2 (2021), 189-221
DOI
10.4310/jdg/1622743139

Comments: v2: We have revised the introduction to better reflect the similarities and differences between our work and related work in the intrinsic case by Marques (Ann. of Math. 2012)

Mean curvature flowSurgery3-manifoldsScalar curvature

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)1Buzano 2017 Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature (Laurent Bessières, Gérard Besson, Sylvain Maillot et al., 2017)2Bessières 2017 Ancient low entropy flows, mean convex neighborhoods, and uniqueness (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2018)7Choi 2018 Constrained deformations of positive scalar curvature metrics (Alessandro Carlotto and Chao Li, 2019)6Carlotto 2019 Classification of noncollapsed translators in ℝ⁴ (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2021)4Choi 2021 Free boundary flow with surgery (Robert Haslhofer, 2023)8Haslhofer 2023 Mean Curvature Flow and Heegaard Surfaces in Lens Spaces (Reto Buzano and Sylvain Maillot, 2023)3Buzano 2023 A gradient estimate for the linearized translator equation (Kyeongsu Choi, Robert Haslhofer and Or Hershkovits, 2025)5Choi 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 tori

    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. Deforming 3-manifolds of bounded geometry and uniformly positive scalar curvature

    Laurent Bessières, Gérard Besson, Sylvain Maillot et al.

    Topics Surgery, 3-manifolds, Scalar curvature · Shared terms “moduli space”, “positive scalar”, “path connected”

  3. Mean Curvature Flow and Heegaard Surfaces in Lens Spaces

    Reto Buzano and Sylvain MaillotWider flows

    Topics 3-manifolds · Same author Reto Buzano · Shared terms “moduli space”, “path connected”

  4. Classification of noncollapsed translators in

    Kyeongsu Choi, Robert Haslhofer and Or HershkovitsWider flows

    Topics 3-manifolds · Same authors Robert Haslhofer, Or Hershkovits

  5. 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”

  6. Constrained deformations of positive scalar curvature metrics

    Alessandro Carlotto and Chao Li

    Topics Surgery, 3-manifolds, Scalar curvature · Shared terms “moduli space”, “positive scalar”, “path connected”

  7. 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

  8. Free boundary flow with surgery

    Robert HaslhoferWider flows

    Topics Surgery · Same author Robert Haslhofer

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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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.