Paper · Jun 6, 2018

Combinatorial Calabi flow with surgery on surfaces

Xiang Zhu, Xu Xu

math.GTWider flowsarXiv:1806.02166v2
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Abstract

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo. To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu and Gu-Luo-Sun-Wu, we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.

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Submitted
Jun 6, 2018
Last revised
Jan 27, 2020 (v2)
Subjects
math.GT, math.DG
Journal
Calc. Var. Partial Differential Equations 58 (2019), no. 6, Art. 195, 20 pp
Yamabe flowOther curvature flowsSurgerySurfacesKähler geometryLong-time existence & convergenceDiscrete & graph Ricci

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 2016, 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 On the deformation of discrete conformal factors on surfaces (Huabin Ge and Wenshuai Jiang, 2016)4Ge 2016 Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I (Xu Xu, 2018)5Xu 2018 Combinatorial p-th Calabi flows on surfaces (Aijin Lin and Xiaoxiao Zhang, 2018)6Lin 2018 Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II (Xu Xu and Chao Zheng, 2021)1Xu 2021 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)3Wu 2021 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)2Luo 2021 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)7Xu 2021 The existence of inversive distance circle packing on polyhedral surface (Xiang Zhu, 2023)8Zhu 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

    Xu Xu and Chao ZhengWider flows

    Topics Surgery, Yamabe flow, Surfaces · Shared terms “Luo Sun”, “combinatorial”, “Calabi flow”

  2. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows

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

  3. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu

    Topics Surgery, Yamabe flow, Discrete & graph Ricci · Shared terms “combinatorial Calabi”, “Calabi flow”, “discrete conformal”

  4. On the deformation of discrete conformal factors on surfaces

    Huabin Ge and Wenshuai JiangWider flows

    Topics Surgery, Yamabe flow, Discrete & graph Ricci · Shared terms “discrete conformal”, “Luo”, “finite number”

  5. Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I

    Xu XuWider flows

    Topics Surgery, Yamabe flow, Discrete & graph Ricci · Shared terms “Luo Sun”, “polyhedral”, “Calabi flow”

  6. Combinatorial -th Calabi flows on surfaces

    Aijin Lin and Xiaoxiao ZhangWider flows

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

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

    Xu Xu

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

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

    Xiang Zhu

    Topics Surgery, Discrete & graph Ricci · Same author Xiang Zhu · Shared terms “polyhedral”, “discrete conformal”, “flipping”

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric inside its conformal class at a rate set by scalar curvature. It is a parabolic approach to the Yamabe problem… 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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 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. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

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

  3. 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.