Paper · Sep 8, 2020

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

Ke Feng, Huabin Ge, Bobo Hua

math.DGarXiv:2009.03731v2
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Abstract

We prove that for a compact 3-manifold M with boundary admitting an ideal triangulation T with valence at least 10 at all edges, there exists a unique complete hyperbolic metric with totally geodesic boundary, so that T is isotopic to a geometric decomposition of M. Our approach is to use a variant of the combinatorial Ricci flow introduced by Luo [Luo05] for pseudo 3-manifolds. In this case, we prove that the extended Ricci flow converges to the hyperbolic metric exponentially fast.

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Submitted
Sep 8, 2020
Last revised
Sep 20, 2020 (v2)
Subjects
math.DG, math.AT, math.GT, math.MG
Journal
Geom. Topol. 26 (2022) 1349-1384
DOI
10.2140/gt.2022.26.1349

Comments: 31 pages. This is a revision of the paper

Ricci flow3-manifoldsLong-time existence & convergenceDiscrete & 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 3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry (Huabin Ge and Bobo Hua, 2018)7Ge 2018 Circle patterns on surfaces of finite topological type (Huabin Ge, Bobo Hua and Ze Zhou, 2019)8Ge 2019 Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)1Feng 2020 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)5Ge 2023 Hyperbolization and geometric decomposition of a class of 3-manifolds (Ke Feng, Huabin Ge and Yunpeng Meng, 2025)2Feng 2025 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)6Ge 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)3Ge 2025 Combinatorial Ricci Flows and Hyperbolic Structures on a Class of Compact 3-Manifolds with Boundary (Xinrong Zhao, 2026)4Zhao 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 flows with applications to the hyperbolization of cusped 3-manifolds

    Ke Feng, Huabin Ge and Bobo Hua

    Topics Discrete & graph Ricci, 3-manifolds, Long-time existence & convergence · Same authors Ke Feng, Huabin Ge, Bobo Hua · Shared terms “hyperbolization”, “combinatorial Ricci”, “extended Ricci”

  2. Hyperbolization and geometric decomposition of a class of 3-manifolds

    Ke Feng, Huabin Ge and Yunpeng Meng

    Topics Discrete & graph Ricci, 3-manifolds · Same authors Ke Feng, Huabin Ge · Shared terms “hyperbolization”, “geometric decomposition”, “ideal triangulation”

  3. 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 · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial Ricci”

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

    Xinrong Zhao

    Topics Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial Ricci”, “hyperbolic metric”, “ideal triangulation”

  5. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial”, “exponentially fast”, “Luo”

  6. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial Ricci”, “triangulation”

  7. 3-dimensional Combinatorial Yamabe Flow in Hyperbolic Background Geometry

    Huabin Ge and Bobo HuaWider flows

    Topics Discrete & graph Ricci, 3-manifolds, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial”, “extended”, “exponentially fast”

  8. Circle patterns on surfaces of finite topological type

    Huabin Ge, Bobo Hua and Ze Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same authors Huabin Ge, Bobo Hua · Shared terms “combinatorial Ricci”, “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.

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) 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) 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) Discrete & graph Ricci Curvature and Ricci flow for graphs, networks and triangulated surfaces. It runs from Chow–Luo's combinatorial flow on circle packings to Ollivier's… Start with: Combinatorial Ricci Flows on Surfaces (Chow et al., 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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  3. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.