Paper · Aug 15, 2019

On the long-time behavior of immortal Ricci flows

Shaosai Huang

math.DGarXiv:1908.05410
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Abstract

For an immortal Ricci flow on an -dimensional closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a Riemannian orbifold; (2) if the flow is type-III with diameter growth controlled by , then any blowdown limit is an -dimensional negative Einstein manifold, provided that Feldman-Ilmanen-Ni's -functional satisfies .

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Submitted
Aug 15, 2019
Subjects
math.DG

Comments: 44 pages

Ricci flowEinstein metricsLong-time existence & convergence

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 Rigidity of the first Betti number via Ricci flow smoothing (Shaosai Huang and Bing Wang, 2020)8Huang 2020 Ricci flow smoothing for locally collapsing manifolds (Shaosai Huang and Bing Wang, 2020)6Huang 2020 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)5Araujo 2023 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)7Tosatti 2024 Orbifold singularity formation along ancient and immortal Ricci flows (Alix Deruelle and Tristan Ozuch, 2024)4Deruelle 2024 Long-time behavior of Ricci flow on some complex surfaces (John Lott, 2025)2Lott 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)3Krishnan 2026 Long-Time Behavior of the α-Ricci Flow on the 2-Torus (Yifan Guo and Xiaokang Wang, 2026)1Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang

    Topics Long-time existence & convergence · Shared terms “time behavior”, “Type III”, “sequence”

  2. Long-time behavior of Ricci flow on some complex surfaces

    John Lott

    Topics Long-time existence & convergence · Shared terms “time behavior”

  3. On the long time behavior of ancient homogeneous Ricci flows

    Anusha M. Krishnan and Francesco Pediconi

    Topics Long-time existence & convergence · Shared terms “time behavior”, “diameter”, “sequence”

  4. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch

    Topics Long-time existence & convergence · Shared terms “immortal Ricci”, “orbifold”

  5. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo

    Topics Long-time existence & convergence · Shared terms “time behavior”

  6. Ricci flow smoothing for locally collapsing manifolds

    Shaosai Huang and Bing Wang

    Topics Ricci flow · Same author Shaosai Huang · Shared terms “Riemannian orbifold”, “controlled”

  7. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti

    Topics Long-time existence & convergence · Shared terms “immortal”

  8. Rigidity of the first Betti number via Ricci flow smoothing

    Shaosai Huang and Bing Wang

    Topics Ricci flow · Same author Shaosai Huang · Shared terms “closed manifold”, “diameter”, “uniformly bounded”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.