Paper · Nov 7, 2017

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

Laurent Bessières, Gérard Besson, Sylvain Maillot, Fernando Coda Marques

math.DGarXiv:1711.02457
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Abstract

We prove that the moduli space of complete Riemannian metrics of bounded geometry and uniformly positive scalar curvature on an orientable 3-manifold is path-connected. This generalizes the main result of the fourth author [Mar12] in the compact case. The proof uses Ricci flow with surgery as well as arguments involving performing infinite connected sums with control on the geometry.

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Submitted
Nov 7, 2017
Subjects
math.DG
Ricci 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 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)5Kleiner 2006 The moduli space of two-convex embedded spheres (Reto Buzano, Robert Haslhofer and Or Hershkovits, 2016)1Buzano 2016 The moduli space of two-convex embedded tori (Reto Buzano, Robert Haslhofer and Or Hershkovits, 2017)7Buzano 2017 Constrained deformations of positive scalar curvature metrics (Alessandro Carlotto and Chao Li, 2019)4Carlotto 2019 Area, Scalar Curvature, and Hyperbolic 3-Manifolds (Ben Lowe, 2021)6Lowe 2021 Open manifolds with uniformly positive isotropic curvature (Hong Huang, 2023)3Huang 2023 Mean Curvature Flow and Heegaard Surfaces in Lens Spaces (Reto Buzano and Sylvain Maillot, 2023)8Buzano 2023 3-Manifolds with positive scalar curvature and bounded geometry (Otis Chodosh, Yi Lai and Kai Xu, 2025)2Chodosh 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, 3-manifolds, Scalar curvature · Shared terms “moduli space”, “positive scalar”, “path connected”

  2. 3-Manifolds with positive scalar curvature and bounded geometry

    Otis Chodosh, Yi Lai and Kai XuWider flows

    Topics 3-manifolds, Scalar curvature · Shared terms “bounded geometry”, “positive scalar”

  3. Open manifolds with uniformly positive isotropic curvature

    Hong Huang

    Topics Surgery, Scalar curvature · Shared terms “uniformly positive”, “connected sum”, “infinite”

  4. Constrained deformations of positive scalar curvature metrics

    Alessandro Carlotto and Chao Li

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

  5. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon

    Topics Surgery, 3-manifolds

  6. Area, Scalar Curvature, and Hyperbolic 3-Manifolds

    Ben Lowe

    Topics Surgery, 3-manifolds, Scalar curvature

  7. The moduli space of two-convex embedded tori

    Reto Buzano, Robert Haslhofer and Or HershkovitsWider flows

    Topics Surgery · Shared terms “moduli space”, “path connected”

  8. Mean Curvature Flow and Heegaard Surfaces in Lens Spaces

    Reto Buzano and Sylvain MaillotWider flows

    Topics 3-manifolds · Same author Sylvain Maillot · Shared terms “path connected”, “moduli space”, “orientable”

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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.