Paper · Jul 29, 2021

Fractional combinatorial Calabi flow on surfaces

Tianqi Wu, Xu Xu

math.GTarXiv:2107.14102
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Abstract

Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for Thurston's circle packings, Luo's combinatorial Yamabe flow for vertex scaling and the combinatorial Calabi flow for discrete conformal structures on surfaces. For Thurston's Euclidean and hyperbolic circle packings on triangulated surfaces, we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow. For vertex scalings on polyhedral surfaces, we do surgery on the fractional combinatorial Calabi flow by edge flipping under the Delaunay condition to handle the potential singularities along the flow. Using the discrete conformal theory established by Gu et al., we prove the longtime existence and global convergence of the fractional combinatorial Calabi flow with surgery.

Comments: 30 pages, 0 figures

Ricci flowYamabe flowOther curvature flowsSingularitiesSurgeryGeometrizationSurfacesKähler geometryLong-time existence & convergenceDiscrete & graph Ricci

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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 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)3Luo 2021 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)4Xu 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)6Xu 2022 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)2Xu 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)7Ba 2023 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)8Zhu 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)1Xu 2023 Discrete conformal structures on surfaces with boundary (III) – Deformation (Xu Xu and Chao Zheng, 2025)5Xu 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Deformation of discrete conformal structures on surfaces

    Xu Xu PDF

    Topics Geometrization, Yamabe flow, Discrete & graph Ricci · Shared terms “discrete conformal”, “conformal structures”, “Luo combinatorial”

  2. Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces

    Xu Xu and Chao Zheng PDF

    Topics Geometrization, Surgery, Discrete & graph Ricci · Shared terms “combinatorial Calabi”, “circle packings”, “fractional combinatorial”

  3. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Yamabe flow, Discrete & graph Ricci, Surfaces · Shared terms “combinatorial Calabi”, “fractional combinatorial”, “Calabi flow”

  4. A new class of discrete conformal structures on surfaces with boundary

    Xu Xu PDF

    Topics Yamabe flow, Discrete & graph Ricci, Surfaces · Shared terms “discrete conformal”, “conformal structures”, “Luo combinatorial”

  5. 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 “discrete conformal”, “conformal structures”, “combinatorial Calabi”

  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 “circle packings”, “combinatorial”, “triangulated surfaces”

  7. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Other curvature flows · Shared terms “circle packings”, “combinatorial Calabi”, “fractional combinatorial”

  8. The existence of inversive distance circle packing on polyhedral surface

    Xiang Zhu PDF

    Topics Surgery, Discrete & graph Ricci, Singularities · Shared terms “discrete conformal”, “circle packings”, “polyhedral 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) Yamabe flow Wider flows The Yamabe flow deforms a metric within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

In the guidebook

  1. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).