Paper · Mar 30, 2021

Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II

Xu Xu, Chao Zheng

math.GTWider flowsarXiv:2103.16077v2
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Abstract

This paper investigates the combinatorial -curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial -curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial -Yamabe flow and combinatorial -Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial -curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu, we prove the longtime existence and convergence of combinatorial -Yamabe flow and combinatorial -Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial -curvatures.

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Submitted
Mar 30, 2021
Last revised
Oct 1, 2021 (v2)
Subjects
math.GT, math.DG

Comments: 31 pages, 0 figure, to appear in Trans. Amer. Math. Soc

Yamabe flowOther curvature flowsSingularitiesSurgerySurfacesKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability

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 Combinatorial Calabi flow with surgery on surfaces (Xiang Zhu and Xu Xu, 2018)4Zhu 2018 Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I (Xu Xu, 2018)2Xu 2018 Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces (Xu Xu and Chao Zheng, 2021)3Xu 2021 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)6Wu 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)8Xu 2022 Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces (Xu Xu and Chao Zheng, 2023)5Xu 2023 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)7Ba 2023 A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces (Xu Xu and Chao Zheng, 2024)1Xu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A discrete uniformization theorem for decorated piecewise hyperbolic metrics on surfaces

    Xu Xu and Chao Zheng

    Topics Surgery, Surfaces, Kähler–Einstein & K-stability · Same author Chao Zheng · Shared terms “discrete uniformization”, “piecewise hyperbolic”, “uniformization theorem”

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

    Xu XuWider flows

    Topics Surgery, Yamabe flow, Surfaces · Shared terms “parameterized discrete”, “discrete uniformization”, “uniformization theorem”

  3. Parameterized combinatorial curvatures and parameterized combinatorial curvature flows for discrete conformal structures on polyhedral surfaces

    Xu Xu and Chao Zheng

    Topics Yamabe flow, Surfaces, Long-time existence & convergence · Same author Chao Zheng · Shared terms “polyhedral surfaces”, “parameterized”, “combinatorial curvature”

  4. Combinatorial Calabi flow with surgery on surfaces

    Xiang Zhu and Xu XuWider flows

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

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

    Xu Xu and Chao Zheng

    Topics Surgery, Surfaces, Kähler–Einstein & K-stability · Same author Chao Zheng · Shared terms “combinatorial curvature”, “prescribed combinatorial”, “piecewise”

  6. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu

    Topics Surgery, Yamabe flow, Surfaces · Shared terms “vertex scaling”, “combinatorial”, “Calabi flow”

  7. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same author Chao Zheng · Shared terms “combinatorial curvature”, “Calabi flow”, “effective algorithms”

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

    Xu Xu and Chao Zheng

    Topics Surfaces, Long-time existence & convergence, Other curvature flows · Same author Chao Zheng · Shared terms “combinatorial curvature”, “hyperbolic metrics”, “Guo Luo”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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).