Paper · May 7, 2026

Notes on harmonic-Ricci flow on surface

Xiang-Zhi Cao

math.DGarXiv:2605.05599
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Abstract

In this note, we want to establish several formulas about functionals along harmonic Ricci flow on surface with boundary

Ricci flowSurfaces

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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 Ricci flow on surfaces along the standard lightcone in the 3+1-Minkowski spacetime (Markus Wolff, 2022)8Wolff 2022 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)7Topping 2023 Determinants of pseudo-laplacians and ζ^((rm reg))(1) for spinor bundles over Riemann surfaces (Alexey Kokotov and Dmitrii Korikov, 2024)5Kokotov 2024 A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric (Gang Li, 2025)4Li 2025 Monotonicity of the Liouville entropy along the Ricci flow on surfaces (Karen Butt, Alena Erchenko, Tristan Humbert et al., 2025)6Butt 2025 Path Connectivity of Anosov Metrics on Surfaces (Guilherme Brandão Guglielmo and R. Ruggiero, 2026)3Guglielmo 2026 Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow (Tianyin Ren and Chong Song, 2026)1Ren 2026 Variations of the harmonic entropy and linear drift on negatively curved surfaces (Karen Butt, Alena Erchenko and Tristan Humbert, 2026)2Butt 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Correspondence between Mean Curvature Flow and Harmonic-Ricci Flow

    Tianyin Ren and Chong Song PDF

    Topics Ricci flow · Shared terms “harmonic Ricci”

  2. Variations of the harmonic entropy and linear drift on negatively curved surfaces

    Karen Butt, Alena Erchenko and Tristan Humbert PDF

    Topics Surfaces

  3. Path Connectivity of Anosov Metrics on Surfaces

    Guilherme Brandão Guglielmo and R. Ruggiero PDF

    Topics Surfaces

  4. A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

    Gang Li PDF

    Topics Surfaces

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

    Alexey Kokotov and Dmitrii Korikov PDF

    Topics Surfaces

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

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

    Topics Surfaces

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

    Peter M. Topping and Hao Yin PDF

    Topics Surfaces

  8. Ricci flow on surfaces along the standard lightcone in the -Minkowski spacetime

    Markus Wolff PDF

    Topics Surfaces

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)

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.