Paper · Mar 1, 2024

On the Hamilton-Lott conjecture in higher dimensions

Alix Deruelle, Felix Schulze, Miles Simon

math.DGarXiv:2403.00708v2
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Abstract

We study -dimensional Ricci flows with non-negative Ricci curvature where the curvature is pointwise controlled by the scalar curvature and bounded by , starting at metric cones which are Reifenberg outside the tip. We show that any such flow behaves like a self-similar solution up to an exponential error in time. As an application, we show that smooth -dimensional complete non-compact Riemannian manifolds which are uniformly PIC1-pinched, with positive asymptotic volume ratio, are Euclidean. This confirms a higher dimensional version of a conjecture of Hamilton and Lott under the assumption of non-collapsing. It also yields a new and more direct proof of the original conjecture of Hamilton and Lott in three dimensions.

Comments: 34 pages, introduction slightly extended and references updated

Ricci flow(Non)collapsingSphere theoremsScalar curvatureWeak & singular flowsBoundaries & non-compact

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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 2021, 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 Ricci flow of W^(2,2)-metrics in four dimensions (Tobias Lamm and Miles Simon, 2021)5Lamm 2021 Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds (Alix Deruelle, Felix Schulze and Miles Simon, 2022)1Deruelle 2022 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)7Lynch 2022 Manifolds with PIC1 pinched curvature (Man-Chun Lee and Peter M. Topping, 2022)4Lee 2022 Expanding Ricci solitons coming out of weakly PIC1 metric cones (Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey, 2024)6Chan 2024 Volume estimates and convergence results for solutions to Ricci flow with L^(p) bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)3Liu 2024 PIC1 pinched manifolds are flat or compact (Alix Deruelle, Man-Chun Lee, Felix Schulze et al., 2026)2Deruelle 2026 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)8Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

    Alix Deruelle, Felix Schulze and Miles Simon PDF

    Topics Weak & singular flows · Same authors Alix Deruelle, Felix Schulze, Miles Simon · Shared terms “dimensional Ricci”, “Lott”, “Reifenberg”

  2. PIC1 pinched manifolds are flat or compact

    Alix Deruelle, Man-Chun Lee, Felix Schulze et al. PDF

    Topics Sphere theorems, Boundaries & non-compact · Same authors Alix Deruelle, Felix Schulze, Miles Simon · Shared terms “PIC”, “dimensional complete”, “Hamilton”

  3. Volume estimates and convergence results for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon PDF

    Topics (Non)collapsing, Scalar curvature · Same author Miles Simon · Shared terms “dimensional Ricci”

  4. Manifolds with PIC1 pinched curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Sphere theorems, Boundaries & non-compact · Shared terms “Lott”, “PIC”, “dimensional complete”

  5. Ricci flow of -metrics in four dimensions

    Tobias Lamm and Miles Simon PDF

    Topics Weak & singular flows, Scalar curvature, Boundaries & non-compact · Same author Miles Simon

  6. Expanding Ricci solitons coming out of weakly PIC1 metric cones

    Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey PDF

    Topics Sphere theorems, Weak & singular flows · Shared terms “metric cones”, “PIC”, “Euclidean”

  7. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego PDF

    Topics Sphere theorems, (Non)collapsing, Boundaries & non-compact · Shared terms “higher dimensions”, “uniformly PIC”, “Euclidean”

  8. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon PDF

    Topics Scalar curvature, Boundaries & non-compact · Same author Miles Simon · Shared terms “Euclidean”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) 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. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.