Paper · Mar 10, 2022

Ricci Flow and Gromov Almost Flat Manifolds

Eric Chen, Guofang Wei, Rugang Ye

math.DGarXiv:2203.05107
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Abstract

We employ the Ricci flow to derive a new theorem about Gromov almost flat manifolds, which generalizes and strengthens the celebrated Gromov–Ruh Theorem. In our theorem, the condition in the Gromov–Ruh Theorem is replaced by the substantially weaker condition .

Ricci flow

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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 Some aspects of Ricci flow on the 4-sphere (Sun-Yung Alice Chang and Eric Chen, 2021)4Chang 2021 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)8Podestà 2022 The Yamabe flow on asymptotically Euclidean manifolds with nonpositive Yamabe constant (Gilles Carron, Eric Chen and Yi Wang, 2022)3Carron 2022 Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature (Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al., 2022)6Khan 2022 Collapsed manifolds with local Ricci bounded covering geometry (Xiaochun Rong, 2022)2Rong 2022 Gap Theorem on Riemannian manifolds using Ricci flow (Pak-Yeung Chan and Man-Chun Lee, 2023)1Chan 2023 A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity (Richard H. Bamler, Eric Chen and Ronan J. Conlon, 2026)7Bamler 2026 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)5Bang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gap Theorem on Riemannian manifolds using Ricci flow

    Pak-Yeung Chan and Man-Chun Lee PDF

    Topics Ricci flow · Shared terms “Ruh Theorem”, “celebrated Gromov”, “strengthens”

  2. Collapsed manifolds with local Ricci bounded covering geometry

    Xiaochun Rong PDF

    Topics Ricci flow · Shared terms “almost flat”, “flat manifolds”, “Gromov”

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

    Gilles Carron, Eric Chen and Yi WangWider flows PDF

    Same author Eric Chen · Shared terms “flat manifolds”

  4. Some aspects of Ricci flow on the 4-sphere

    Sun-Yung Alice Chang and Eric Chen PDF

    Topics Ricci flow · Same author Eric Chen

  5. Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

    Kyeongho Bang, Eric Chen and Wenkui Du PDF

    Topics Ricci flow · Same author Eric Chen

  6. Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

    Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al. PDF

    Topics Ricci flow · Same author Guofang Wei

  7. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon PDF

    Same author Eric Chen

  8. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero PDF

    Topics Ricci flow · Shared terms “flat manifolds”

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)

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.