Paper · Sep 12, 2020

Combinatorial Ricci flows with applications to the hyperbolization of cusped 3-manifolds

Ke Feng, Huabin Ge, Bobo Hua

math.DGarXiv:2009.05842
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Abstract

In this paper, we adopt combinatorial Ricci curvature flow methods to study the existence of cusped hyperbolic structure on 3-manifolds with torus boundary. For general pseudo 3-manifolds, we prove the long-time existence and the uniqueness for the extended Ricci flow for decorated hyperbolic polyhedral metrics. We prove that the extended Ricci flow converges to a decorated hyperbolic polyhedral metric if and only if there exists a decorated hyperbolic polyhedral metric of zero Ricci curvature. If it is the case, the flow converges exponentially fast. These results apply for cusped hyperbolic structure on 3-manifolds via ideal triangulation.

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Submitted
Sep 12, 2020
Subjects
math.DG, math.GT, math.MG
Ricci flow3-manifoldsStability & uniquenessLong-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 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 and the hyperbolization of a class of compact 3-manifolds (Ke Feng, Huabin Ge and Bobo Hua, 2020)1Feng 2020 Combinatorial Ricci flow on cusped 3-manifolds (Xu Xu, 2020)6Xu 2020 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)4Ge 2023 Hyperbolization and geometric decomposition of a class of 3-manifolds (Ke Feng, Huabin Ge and Yunpeng Meng, 2025)3Feng 2025 Combinatorial Ricci flows on infinite disk triangulations (Huabin Ge, Bobo Hua and Puchun Zhou, 2025)5Ge 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)2Ge 2025
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 and the hyperbolization of a class of compact 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. 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”

  3. 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”, “ideal triangulation”, “torus boundary”

  4. 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”, “flow converges”, “exponentially fast”

  5. Combinatorial Ricci flows on infinite disk triangulations

    Huabin Ge, Bobo Hua and Puchun Zhou

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

  6. Combinatorial Ricci flow on cusped 3-manifolds

    Xu Xu

    Topics Discrete & graph Ricci, 3-manifolds · Shared terms “combinatorial Ricci”, “decorated hyperbolic”, “hyperbolic polyhedral”

  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”, “flow converges”

  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”

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) 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. 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.