Paper · Jan 21, 2026

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

Xinrong Zhao

math.GTarXiv:2601.15174v3
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Abstract

In this paper, we study a combinatorial Ricci flow on closed pseudo -manifolds . We prove that if every edge in the triangulation has valence at least , then the combinatorial Ricci flow converges exponentially fast to the unique zero-curvature hyper-ideal metric. As a consequence, for any compact -manifold with boundary admitting an ideal triangulation whose edges all have valence at least , there exists a unique complete hyperbolic metric with totally geodesic boundary on such that is isotopic to a geometric decomposition of . This provides a partial solution to the conjecture of Costantino, Frigerio, Martelli and Petronio, and hence an affirmative answer to Thurston's geometric ideal triangulation conjecture for such manifolds. Moreover, we obtain explicit upper and lower bounds for the resulting hyperbolic metric.

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Submitted
Jan 21, 2026
Last revised
Sep 24, 2026 (v3)
Subjects
math.GT

Comments: 28 pages, 2 figures

Ricci flowGeometrizationBoundaries & non-compactLong-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 2021, 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 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)8Ge 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)7Xu 2023 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)5Zhang 2025 Combinatorial Ricci Flow and Thurston's Triangulation Conjecture (Feng Ke and Ge Huabin, 2025)3Ke 2025 Hyperbolization and geometric decomposition of a class of 3-manifolds (Ke Feng, Huabin Ge and Yunpeng Meng, 2025)1Feng 2025 A prescribed curvature flow on hyperbolic surfaces with infinite topological type (Xinrong Zhao and Puchun Zhou, 2025)4Zhao 2025 Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow (Huabin Ge, Bobo Hua, Hao Yu et al., 2025)6Ge 2025 Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow (Soichiro Uemura, 2026)2Uemura 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Hyperbolization and geometric decomposition of a class of 3-manifolds

    Ke Feng, Huabin Ge and Yunpeng Meng

    Topics Geometrization, Discrete & graph Ricci, Boundaries & non-compact · Shared terms “ideal triangulation”, “geometric decomposition”, “hyper ideal”

  2. Local Combinatorial Criteria for Geometric Hyper-ideal Triangulations via Combinatorial Ricci Flow

    Soichiro Uemura

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

  3. Combinatorial Ricci Flow and Thurston's Triangulation Conjecture

    Feng Ke and Ge Huabin

    Topics Geometrization, Discrete & graph Ricci · Shared terms “combinatorial Ricci”, “triangulation”, “hyperbolic structures”

  4. A prescribed curvature flow on hyperbolic surfaces with infinite topological type

    Xinrong Zhao and Puchun Zhou

    Topics Discrete & graph Ricci, Boundaries & non-compact · Same author Xinrong Zhao · Shared terms “geodesic boundaries”, “decomposition”

  5. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang

    Topics Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial”, “exponentially fast”, “converges exponentially”

  6. 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, Boundaries & non-compact · Shared terms “combinatorial Ricci”, “affirmative”

  7. Deformation of discrete conformal structures on surfaces

    Xu Xu

    Topics Geometrization, Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial Ricci”, “Thurston”, “exponentially fast”

  8. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou

    Topics Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial”, “exponentially fast”, “converges exponentially”

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) Geometrization Thurston's conjecture: every closed 3-manifold can be cut along spheres and tori into pieces that each carry one of eight model geometries. Start with: Three dimensional manifolds, Kleinian groups and hyperbolic geometry (Thurston, 1982) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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. 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.

  2. 10A.8
    Thurston’s Eight GeometriesBook 10A · Three-Manifolds and Geometrization

    The model geometries, the geometrization conjecture, and why it implies Poincaré.

  3. 10A.9
    Hyperbolic Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    Ideal tetrahedra, Mostow rigidity, and the thick–thin decomposition.