Paper · Sep 5, 2020

Combinatorial Ricci flow on compact 3-manifolds with boundary

Xu Xu

math.GTarXiv:2009.02496
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Abstract

Combinatorial Ricci flow on an ideally triangulated compact 3-manifold with boundary was introduced by Luo as a 3-dimensional analog of Chow-Luo's combinatorial Ricci flow on a triangulated surface and conjectured to find algorithmically the complete hyperbolic metric on the compact 3-manifold with totally geodesic boundary. In this paper, we prove Luo's conjecture affirmatively by extending the combinatorial Ricci flow through the singularities of the flow if the ideally triangulated compact 3-manifold with boundary admits such a metric.

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Submitted
Sep 5, 2020
Subjects
math.GT
Journal
J. Funct. Anal. 283 (2022), no. 5, Paper No. 109556, 19 pp

Comments: 18 pages

Ricci flowSingularities3-manifoldsWeak & singular flowsBoundaries & non-compactDiscrete & graph Ricci

Its neighbourhood

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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 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 Convergence of combinatorial Ricci flows to degenerate circle patterns (Asuka Takatsu, 2018)7Takatsu 2018 Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)2Feng 2020 Combinatorial Ricci flow on cusped 3-manifolds (Xu Xu, 2020)1Xu 2020 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)8Luo 2021 A new class of discrete conformal structures on surfaces with boundary (Xu Xu, 2021)5Xu 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)3Xu 2022 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)4Ge 2025 Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary (Xinrong Zhao, 2026)6Zhao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Combinatorial Ricci flow on cusped 3-manifolds

    Xu Xu

    Topics Discrete & graph Ricci, 3-manifolds, Boundaries & non-compact · Shared terms “combinatorial Ricci”, “complete hyperbolic”, “Luo”

  2. Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, 3-manifolds · Shared terms “combinatorial Ricci”, “hyperbolic metric”, “geodesic boundaries”

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

    Xu Xu and Chao Zheng

    Topics Discrete & graph Ricci · Shared terms “ideally triangulated”, “combinatorial”, “geodesic boundaries”

  4. Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow

    Huabin Ge, Bobo Hua, Hao Yu et al.

    Topics Discrete & graph Ricci, 3-manifolds, Boundaries & non-compact · Shared terms “combinatorial Ricci”

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

    Xu Xu

    Topics Discrete & graph Ricci, 3-manifolds · Shared terms “combinatorial Ricci”, “Luo”, “geodesic boundaries”

  6. Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact -Manifolds with Boundary

    Xinrong Zhao

    Topics Discrete & graph Ricci, Boundaries & non-compact · Shared terms “combinatorial Ricci”, “hyperbolic metric”, “geodesic boundaries”

  7. Convergence of combinatorial Ricci flows to degenerate circle patterns

    Asuka Takatsu

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

  8. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows

    Topics Discrete & graph Ricci · Shared terms “combinatorial”, “ideally triangulated”, “geodesic boundaries”

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) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 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.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.