Paper · Apr 28, 2016

On the deformation of discrete conformal factors on surfaces

Huabin Ge, Wenshuai Jiang

math.GTWider flowsarXiv:1604.08318
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Abstract

In, Feng Luo conjectured that the discrete Yamabe flow will converge to the constant curvature PL-metric after finite number of surgeries on the triangulation. In this paper, we prove that the flow can always be extended (without surgeries) to a solution that converges exponentially fast to the constant curvature PL-metric.

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Submitted
Apr 28, 2016
Subjects
math.GT, math.DG

Comments: 12 pages. arXiv admin note: text overlap with arXiv:1604.08317

Yamabe flowSurgerySurfacesLong-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 inversive distance circle packings (Huabin Ge and Wenshuai Jiang, 2016)5Ge 2016 On the deformation of inversive distance circle packings, II (Huabin Ge and Wenshuai Jiang, 2016)3Ge 2016 On the deformation of ball packings (Huabin Ge, Wenshuai Jiang and Liangming Shen, 2018)1Ge 2018 3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry (Huabin Ge and Bobo Hua, 2018)6Ge 2018 Combinatorial Calabi flow with surgery on surfaces (Xiang Zhu and Xu Xu, 2018)2Zhu 2018 Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I (Xu Xu, 2018)4Xu 2018 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)8Xu 2021 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)7Xu 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the deformation of ball packings

    Huabin Ge, Wenshuai Jiang and Liangming ShenWider flows

    Topics Yamabe flow, Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Wenshuai Jiang · Shared terms “constant curvature”, “triangulation”, “exponentially fast”

  2. Combinatorial Calabi flow with surgery on surfaces

    Xiang Zhu and Xu XuWider flows

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

  3. On the deformation of inversive distance circle packings, II

    Huabin Ge and Wenshuai Jiang

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Wenshuai Jiang · Shared terms “Luo”, “exponentially fast”, “extended”

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

    Xu XuWider flows

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

  5. On the deformation of inversive distance circle packings

    Huabin Ge and Wenshuai Jiang

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Wenshuai Jiang · Shared terms “Luo”, “exponentially fast”, “converges exponentially”

  6. 3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

    Huabin Ge and Bobo HuaWider flows

    Topics Yamabe flow, Discrete & graph Ricci, Long-time existence & convergence · Same author Huabin Ge · Shared terms “extended”, “exponentially fast”, “triangulation”

  7. Deformation of discrete conformal structures on surfaces

    Xu Xu

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

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

    Xu Xu

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

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.