Paper · Mar 5, 2024

Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

Albert Chau, Adam Martens

math.DGarXiv:2403.02564v2Communications in Analysis and Geometry. Volume 34. Issue 1. 2026. Pages 285-333
PDF Abstract page

As on arXiv

Abstract

We prove long-time existence of the Ricci flow starting from complete manifolds with bounded curvature and scale-invariant integral curvature sufficiently pinched with respect to the inverse of its Sobolev constant. Moreover, if the curvature is sub-critical -integrable, this flow converges locally smoothly to a limiting metric on with isometric to the standard flat , which implies topological rigidity of . This generalizes work of Chen, who proved analogous results for asymptotically flat manifolds. We also prove a long-time Ricci flow existence (and likewise topological rigidity) result for unbounded curvature initial data, assuming the initial data is a locally smooth limit of bounded curvature manifolds as described above.

Comments: Published version, in CAG. Minor changes only from previous version

Ricci flowStability & uniquenessBoundaries & non-compactLong-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 Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows (Albert Chau and Adam Martens, 2023)2Chau 2023 On the Isometry Group of Immortal Homogeneous Ricci Flows (Roberto Araujo, 2023)8Araujo 2023 Long-time behavior of awesome homogeneous Ricci flows (Roberto Araujo, 2023)4Araujo 2023 Sharpening a gap theorem: nonnegative Ricci and small curvature concentration (Adam Martens, 2024)3Martens 2024 Removing scalar curvature assumption for Ricci flow smoothing (Adam Martens, 2024)5Martens 2024 Stability of asymptotically conical gradient Kähler-Ricci expanders (Longteng Chen, 2025)6Chen 2025 A note on Ricci flow from small curvature concentration and a Morrey-type condition (Albert Chau and Adam Martens, 2026)1Chau 2026 Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow (Longteng Chen, 2026)7Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A note on Ricci flow from small curvature concentration and a Morrey-type condition

    Albert Chau and Adam Martens PDF

    Topics Long-time existence & convergence, Boundaries & non-compact · Same authors Albert Chau, Adam Martens · Shared terms “invariant integral”, “bounded curvature”, “integral curvature”

  2. Pseudolocality and completeness for nonnegative Ricci curvature limits of 3D singular Ricci flows

    Albert Chau and Adam Martens PDF

    Topics Boundaries & non-compact · Same authors Albert Chau, Adam Martens

  3. Sharpening a gap theorem: nonnegative Ricci and small curvature concentration

    Adam Martens PDF

    Topics Ricci flow · Same author Adam Martens · Shared terms “invariant integral”, “time Ricci”, “integral curvature”

  4. Long-time behavior of awesome homogeneous Ricci flows

    Roberto Araujo PDF

    Topics Long-time existence & convergence, Boundaries & non-compact

  5. Removing scalar curvature assumption for Ricci flow smoothing

    Adam Martens PDF

    Topics Stability & uniqueness · Same author Adam Martens · Shared terms “time existence”

  6. Stability of asymptotically conical gradient Kähler-Ricci expanders

    Longteng Chen PDF

    Topics Long-time existence & convergence, Boundaries & non-compact, Stability & uniqueness · Shared terms “limiting metric”, “flow starting”

  7. Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow

    Longteng Chen PDF

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “initial data”

  8. On the Isometry Group of Immortal Homogeneous Ricci Flows

    Roberto Araujo PDF

    Topics Long-time existence & convergence, Boundaries & non-compact

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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.