Paper · Jan 14, 2021

Gradient inequality and convergence of the normalized Ricci flow

Nikos I. Kavallaris, Takashi Suzuki

math.DGarXiv:2101.05554
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Abstract

We study the problem of convergence of the normalized Ricci flow evolving on a compact manifold without boundary. In we derived, via PDE techniques, global-in-time existence of the classical solution and pre-compactness of the orbit. In this work we show its convergence to steady-states, using a gradient inequality of Łojasiewicz type. We have thus an alternative proof of, but for general manifold and not only for unit sphere. As a byproduct of that approach we also derive the rate of convergence according to this steady-sate being either degenerate or non-degenerate as a critical point of a related energy functional.

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Submitted
Jan 14, 2021
Subjects
math.DG, math.AP
Ricci flowLimit spaces & RCDLong-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 Convergence of Ricci flows with bounded scalar curvature (Richard H. Bamler, 2016)1Bamler 2016 A gradient flow of isometric G₂ structures (Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis, 2019)3Dwivedi 2019 Convergence of Hermitian manifolds and the Type IIB flow (Nikita Klemyatin, 2021)7Klemyatin 2021 Ricci curvature integrals, local functionals, and the Ricci flow (Yuanqing Ma and Bing Wang, 2021)8Ma 2021 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)4Krishnan 2026 Ricci flow with metric torsion on surfaces of positive Euler characteristic (Shubham Dwivedi, 2026)5Dwivedi 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)6Angella 2026 Long-Time Behavior of the α-Ricci Flow on the 2-Torus (Yifan Guo and Xiaokang Wang, 2026)2Guo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of Ricci flows with bounded scalar curvature

    Richard H. Bamler

    Topics Limit spaces & RCD, Long-time existence & convergence · Shared terms “compactness”

  2. Long-Time Behavior of the -Ricci Flow on the 2-Torus

    Yifan Guo and Xiaokang Wang

    Topics Limit spaces & RCD, Long-time existence & convergence

  3. A gradient flow of isometric structures

    Shubham Dwivedi, Panagiotis Gianniotis and Spiro Karigiannis

    Topics Limit spaces & RCD, Long-time existence & convergence · Shared terms “energy functional”, “compactness”

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

    Anusha M. Krishnan and Francesco Pediconi

    Topics Limit spaces & RCD, Long-time existence & convergence · Shared terms “compact manifolds”

  5. Ricci flow with metric torsion on surfaces of positive Euler characteristic

    Shubham Dwivedi

    Topics Long-time existence & convergence · Shared terms “gradient inequality”, “Łojasiewicz”

  6. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa

    Topics Limit spaces & RCD, Long-time existence & convergence

  7. Convergence of Hermitian manifolds and the Type IIB flow

    Nikita Klemyatin

    Topics Limit spaces & RCD · Shared terms “compactness”

  8. Ricci curvature integrals, local functionals, and the Ricci flow

    Yuanqing Ma and Bing Wang

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

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.