Paper · Jan 22, 2025

Long-time behavior of Ricci flow on some complex surfaces

John Lott

math.DGarXiv:2501.12759v2
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Abstract

We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.

Comments: final version

Ricci flow4-manifoldsLong-time existence & convergenceMinimal & CMC surfacesAnalytic minimal model program

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 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)1Araujo 2023 On the long-time limit of the mean curvature flow in closed manifolds (Alexander Mramor and Ao Sun, 2024)8Mramor 2024 Expanding Ricci solitons and Higgs bundles (Ramiro A. Lafuente and Adam Thompson, 2025)6Lafuente 2025 The Dirichlet problem for the minimal surface system on smooth domains (Caiyan Li and Hengyu Zhou, 2025)7Li 2025 On the long time behavior of ancient homogeneous Ricci flows (Anusha M. Krishnan and Francesco Pediconi, 2026)5Krishnan 2026 Convergence of the Chern-Ricci flow on complex minimal surfaces of general type (Haoyuan Sun, 2026)3Sun 2026 Under Ricci flow, a 3-torus goes flat (John Lott, 2026)2Lott 2026 Long-Time Behavior of the α-Ricci Flow on the 2-Torus (Yifan Guo and Xiaokang Wang, 2026)4Guo 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 awesome homogeneous Ricci flows

    Roberto Araujo PDF

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

  2. Under Ricci flow, a 3-torus goes flat

    John Lott PDF

    Topics Long-time existence & convergence · Same author John Lott · Shared terms “time behavior”

  3. Convergence of the Chern-Ricci flow on complex minimal surfaces of general type

    Haoyuan Sun PDF

    Topics Analytic minimal model program, Minimal & CMC surfaces

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

    Yifan Guo and Xiaokang Wang PDF

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

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

    Anusha M. Krishnan and Francesco Pediconi PDF

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

  6. Expanding Ricci solitons and Higgs bundles

    Ramiro A. Lafuente and Adam Thompson PDF

    Topics 4-manifolds, Long-time existence & convergence · Shared terms “time behavior”

  7. The Dirichlet problem for the minimal surface system on smooth domains

    Caiyan Li and Hengyu ZhouWider flows PDF

    Topics Minimal & CMC surfaces, Long-time existence & convergence

  8. On the long-time limit of the mean curvature flow in closed manifolds

    Alexander Mramor and Ao SunWider flows PDF

    Topics Minimal & CMC surfaces · Shared terms “long time”

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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.