Paper · Apr 2, 2024

A Simple Ricci Flow Proof of the Uniformization Theorem

Yucheng Ji

math.DGarXiv:2404.01980v6
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Abstract

In this note, we provide a very simple proof of the uniformization theorem of Riemann surfaces by Ricci flow. The argument builds on a refinement of Hamilton's isoperimetric estimate for the Ricci flow on the two-sphere.

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Submitted
Apr 2, 2024
Last revised
Aug 25, 2024 (v6)
Subjects
math.DG, math.CV

Comments: There is a flaw in the proof of Theorem 3.3 that the A at t=0 and A at t=T cannot be guaranteed to be identical, thus the key inequality fails

Ricci flowSurfacesGeometric inequalities

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 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)7Topping 2021 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)3Topping 2023 On local rigidity theorems with respect to the scalar curvature (Liang Cheng, 2023)8Cheng 2023 A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces (Xu Xu and Chao Zheng, 2024)2Xu 2024 Determinants of pseudo-laplacians and ζ^((rm reg))(1) for spinor bundles over Riemann surfaces (Alexey Kokotov and Dmitrii Korikov, 2024)4Kokotov 2024 Monotonicity of the Cheeger constant under Ricci flow on spheres (Hollis Williams, 2024)1Williams 2024 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)5Ge 2025 Notes on harmonic-Ricci flow on surface (Xiang-Zhi Cao, 2026)6Cao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Monotonicity of the Cheeger constant under Ricci flow on spheres

    Hollis Williams

    Topics Geometric inequalities, Surfaces · Shared terms “isoperimetric”

  2. A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

    Xu Xu and Chao Zheng

    Topics Surfaces · Shared terms “uniformization theorem”

  3. Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

    Peter M. Topping and Hao Yin

    Topics Surfaces · Shared terms “Riemann surface”

  4. Determinants of pseudo-laplacians and for spinor bundles over Riemann surfaces

    Alexey Kokotov and Dmitrii Korikov

    Topics Surfaces · Shared terms “Riemann surface”

  5. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Surfaces · Shared terms “uniformization theorem”

  6. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao

    Topics Surfaces

  7. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin

    Topics Surfaces · Shared terms “Riemann surface”

  8. On local rigidity theorems with respect to the scalar curvature

    Liang Cheng

    Topics Geometric inequalities · Shared terms “isoperimetric”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) Geometric inequalities Sharp inequalities between area, volume and integrals of curvature, with equality only for round spheres. Start with: Mean curvature flow and geometric inequalities (Topping, 1998)

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. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.