Paper · May 30, 2025

Branched -combinatorial Ricci flows on closed surfaces with Euler characteristic

Wenjun Li, Rongyuan Liu, Guohao Chen, Aijin Lin

math.DGarXiv:2505.24762v4
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Abstract

In this paper we introduce the branched -flows on closed surfaces with Euler characteristic . Based on the strict convexity of the branched -potentials, we establish the long time existence and convergence of the solutions to the branched -flows, which generalizes Ge and Xu's main results on the -flows. In addtion, we study the prescribed curvature problems under the relaxed precondition via alternative -flows, establishing admissibility conditions for prescribed curvatures and their exponential convergence to target metrics.

Ricci flowSurfacesLong-time existence & convergenceDiscrete & graph Ricci

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  1. Ricci flow with metric torsion on surfaces of positive Euler characteristic

    Shubham Dwivedi PDF

    Topics Surfaces, Long-time existence & convergence · Shared terms “Euler characteristic”

  2. A Variant Prescribed Curvature Flow on Closed Surfaces with Negative Euler Characteristic

    Franziska Borer, Peter Elbau and Tobias WethWider flows PDF

    Topics Surfaces · Shared terms “Euler characteristic”, “closed surfaces”, “prescribed curvature”

  3. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “prescribed curvature”, “combinatorial Ricci”

  4. Generalized circle patterns on surfaces with cusps

    Zhiwen Xiong and Xu Xu PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “closed surfaces”, “combinatorial Ricci”

  5. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Surfaces · Shared terms “combinatorial Ricci”

  6. Combinatorial curvature flows for generalized circle packings on surfaces with boundary

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “combinatorial”

  7. The character of ideal circle patterns

    Chang Li, Aijin Lin and Liangming Shen PDF

    Topics Discrete & graph Ricci · Same author Aijin Lin · Shared terms “closed surfaces”, “combinatorial Ricci”

  8. Discrete conformal structures on surfaces with boundary (III) – Deformation

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Surfaces, Long-time existence & convergence · Shared terms “combinatorial Ricci”

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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) 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) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 2002)

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. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  3. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.