Paper · Jun 14, 2023

Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

Peter M. Topping, Hao Yin

math.DGarXiv:2306.08398
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Abstract

In previous work we established the existence of a Ricci flow starting with a Riemann surface coupled with a nonatomic Radon measure as a conformal factor. In this paper we prove uniqueness. Combining these two works yields a canonical smoothing of such rough surfaces that also regularises their geometry at infinity.

Ricci flowSurfacesStability & uniqueness

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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 Smoothing a measure on a Riemann surface using Ricci flow (Peter M. Topping and Hao Yin, 2021)1Topping 2021 All two-dimensional expanding Ricci solitons (Luke T. Peachey and Peter M. Topping, 2023)2Peachey 2023 Determinants of pseudo-laplacians and ζ^((rm reg))(1) for spinor bundles over Riemann surfaces (Alexey Kokotov and Dmitrii Korikov, 2024)3Kokotov 2024 A Simple Ricci Flow Proof of the Uniformization Theorem (Yucheng Ji, 2024)7Ji 2024 Monotonicity of the Liouville entropy along the Ricci flow on surfaces (Karen Butt, Alena Erchenko, Tristan Humbert et al., 2025)8Butt 2025 Path Connectivity of Anosov Metrics on Surfaces (Guilherme Brandão Guglielmo and R. Ruggiero, 2026)5Guglielmo 2026 Notes on harmonic-Ricci flow on surface (Xiang-Zhi Cao, 2026)6Cao 2026 Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow (Ping-Hung Lee, 2026)4Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Smoothing a measure on a Riemann surface using Ricci flow

    Peter M. Topping and Hao Yin PDF

    Topics Surfaces · Same authors Peter M. Topping, Hao Yin · Shared terms “Riemann surface”, “Radon measure”, “smoothing”

  2. All two-dimensional expanding Ricci solitons

    Luke T. Peachey and Peter M. Topping PDF

    Topics Surfaces, Stability & uniqueness · Same author Peter M. Topping · Shared terms “nonatomic Radon”, “Radon measure”, “flow starting”

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

    Alexey Kokotov and Dmitrii Korikov PDF

    Topics Surfaces · Shared terms “Riemann surface”

  4. Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

    Ping-Hung Lee PDF

    Topics Surfaces · Shared terms “Riemann surface”

  5. Path Connectivity of Anosov Metrics on Surfaces

    Guilherme Brandão Guglielmo and R. Ruggiero PDF

    Topics Surfaces

  6. Notes on harmonic-Ricci flow on surface

    Xiang-Zhi Cao PDF

    Topics Surfaces

  7. A Simple Ricci Flow Proof of the Uniformization Theorem

    Yucheng Ji PDF

    Topics Surfaces · Shared terms “Riemann surface”

  8. Monotonicity of the Liouville entropy along the Ricci flow on surfaces

    Karen Butt, Alena Erchenko, Tristan Humbert et al. PDF

    Topics Surfaces

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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) 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)

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.