Paper · Oct 13, 2026

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

Yifan Guo, Xiaokang Wang

math.DGarXiv:2610.19273
PDF Abstract page

As on arXiv

Abstract

We study the -Ricci flow , which is a Ricci flow coupled with a harmonic 1-form heat flow, on the 2-torus . We prove global existence for this flow as well as a Type III curvature estimate. The long-time behavior of the -Ricci flow depends on the cohomology class , which is invariant under the flow. If is rational, then as , Gromov–Hausdorff converges to with length where is the primitive length of . On the universal cover, the rescaled flow smoothly converges to an expander on as where is nonzero and constant in spacetime and is flat and expanding in the direction. If is irrational, then there is a sequence such that Gromov–Hausdorff converges to a point. Moreover, on the universal cover, smoothly converges to .

Comments: 49 pages, 2 figures

Ricci flowRicci solitonsLimit 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 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 Time zero regularity of Ricci flow (Man-Chun Lee and Peter M. Topping, 2022)8Lee 2022 Ricci flow from spaces with edge type conical singularities (Lucas Lavoyer, 2023)7Lavoyer 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)4Araujo 2023 Long-time behavior of Ricci flow on some complex surfaces (John Lott, 2025)5Lott 2025 Pluriclosed flow on Oeljeklaus-Toma manifolds (Jeffrey Streets and Xiaokang Wang, 2025)2Streets 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)3Krishnan 2026 Under Ricci flow, a 3-torus goes flat (John Lott, 2026)1Lott 2026 Chern-Ricci flow on Kato surfaces (Daniele Angella and Mauricio Corrêa, 2026)6Angella 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Under Ricci flow, a 3-torus goes flat

    John Lott PDF

    Topics Limit spaces & RCD, Long-time existence & convergence, Ricci solitons · Shared terms “time behavior”, “universal cover”, “Hausdorff”

  2. Pluriclosed flow on Oeljeklaus-Toma manifolds

    Jeffrey Streets and Xiaokang Wang PDF

    Topics Limit spaces & RCD, Long-time existence & convergence · Same author Xiaokang Wang · Shared terms “global existence”, “Hausdorff”, “Gromov”

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

    Anusha M. Krishnan and Francesco Pediconi PDF

    Topics Limit spaces & RCD, Long-time existence & convergence, Ricci solitons · Shared terms “time behavior”, “sequence”

  4. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

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

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

    John Lott PDF

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

  6. Chern-Ricci flow on Kato surfaces

    Daniele Angella and Mauricio Corrêa PDF

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

  7. Ricci flow from spaces with edge type conical singularities

    Lucas Lavoyer PDF

    Topics Limit spaces & RCD, Ricci solitons · Shared terms “curvature estimates”, “Hausdorff”, “Gromov”

  8. Time zero regularity of Ricci flow

    Man-Chun Lee and Peter M. Topping PDF

    Topics Limit spaces & RCD · Shared terms “curvature estimates”, “Hausdorff”, “Gromov”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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.