Paper · Jul 30, 2019

Convergence of the Ricci flow on asymptotically flat manifolds with integral curvature pinching

Eric Chen

math.DGarXiv:1907.13189
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Abstract

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to .

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Submitted
Jul 30, 2019
Subjects
math.DG, math.AP

Comments: 33 pages, comments welcome

Ricci flowCurvature pinchingBoundaries & 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 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 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)6Koike 2017 The Yamabe flow on asymptotically flat manifolds (Eric Chen and Yi Wang, 2021)2Chen 2021 Small curvature concentration and Ricci flow smoothing (Pak-Yeung Chan, Eric Chen and Man-Chun Lee, 2021)4Chan 2021 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)5Lee 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)3Carron 2022 Long-time Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature (Albert Chau and Adam Martens, 2024)1Chau 2024 Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching (Jian Ge, 2026)7Ge 2026 Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow (Tsz-Kiu Aaron Chow, 2026)8Chow 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 Ricci flow existence and topological rigidity from manifolds with pinched scale-invariant integral curvature

    Albert Chau and Adam Martens

    Topics Long-time existence & convergence, Boundaries & non-compact · Shared terms “integral curvature”, “invariant integral”, “scale invariant”

  2. The Yamabe flow on asymptotically flat manifolds

    Eric Chen and Yi WangWider flows

    Topics Boundaries & non-compact · Same author Eric Chen · Shared terms “asymptotically flat”, “flat manifolds”, “flow starting”

  3. The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant

    Gilles Carron, Eric Chen and Yi WangWider flows

    Topics Boundaries & non-compact · Same author Eric Chen · Shared terms “flat manifolds”, “asymptotically flat”, “flow starting”

  4. Small curvature concentration and Ricci flow smoothing

    Pak-Yeung Chan, Eric Chen and Man-Chun Lee

    Topics Boundaries & non-compact · Same author Eric Chen · Shared terms “integral curvature”, “invariant integral”, “scale invariant”

  5. Three-manifolds with non-negatively pinched Ricci curvature

    Man-Chun Lee and Peter M. Topping

    Topics Curvature pinching, Boundaries & non-compact · Shared terms “scale invariant”, “positive time”

  6. Mean curvature flow for pinched submanifolds in rank one symmetric spaces

    Naoyuki Koike, Yoshiyuki Mizumura and Nana UenoyamaWider flows

    Topics Curvature pinching, Long-time existence & convergence, Boundaries & non-compact · Shared terms “flow starting”

  7. Sharp Weitzenböck and PIC2 Estimates from Sectional-Scalar Curvature Pinching

    Jian Ge

    Topics Curvature pinching · Shared terms “Euclidean”

  8. Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

    Tsz-Kiu Aaron Chow

    Topics Curvature pinching · Shared terms “Euclidean space”

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  2. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.