Paper · Oct 19, 2026

Combinatorial Calabi flows on infinitely triangulated surfaces

Yuerong Bian, Xu Xu

math.DGWider flowsarXiv:2610.25745
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Abstract

We study the combinatorial Calabi flow associated with vertex scalings on infinitely triangulated surfaces with piecewise Euclidean and piecewise hyperbolic metrics. Under the assumptions of uniformly bounded vertex degree and an -uniformly nondegenerate initial metric, we establish short-time existence of smooth solutions and prove uniqueness among solutions that are continuous in the norm.

Other curvature flowsKähler geometryStability & uniquenessHarmonic maps & DeTurck trickDiscrete & graph Ricci

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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 Fractional combinatorial Calabi flow on surfaces (Tianqi Wu and Xu Xu, 2021)6Wu 2021 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)2Luo 2021 Combinatorial curvature flows for generalized circle packings on surfaces with boundary (Xu Xu and Chao Zheng, 2022)5Xu 2022 Combinatorial curvature flows for generalized hyperbolic circle packings (Te Ba and Chao Zheng, 2023)8Ba 2023 Deformation of discrete conformal structures on surfaces (Xu Xu, 2023)7Xu 2023 Combinatorial Calabi flows with ideal circle patterns (Xiaoxiao Zhang, 2025)3Zhang 2025 The existence and uniqueness of infinite combinatorial Yamabe flows (Bohao Ji, 2025)4Ji 2025 Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces (Yuerong Bian and Xu Xu, 2026)1Bian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

    Yuerong Bian and Xu XuWider flows PDF

    Topics Harmonic maps & DeTurck trick, Discrete & graph Ricci, Stability & uniqueness · Same author Yuerong Bian · Shared terms “infinitely triangulated”, “triangulated surfaces”, “combinatorial”

  2. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “combinatorial Calabi”, “Calabi flow”, “triangulated surfaces”

  3. Combinatorial Calabi flows with ideal circle patterns

    Xiaoxiao Zhang PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “combinatorial Calabi”, “Calabi flow”, “Euclidean”

  4. The existence and uniqueness of infinite combinatorial Yamabe flows

    Bohao JiWider flows PDF

    Topics Harmonic maps & DeTurck trick, Discrete & graph Ricci, Stability & uniqueness · Shared terms “triangulated surfaces”, “combinatorial”, “time existence”

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

    Xu Xu and Chao Zheng PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “triangulated surfaces”, “combinatorial”, “hyperbolic metrics”

  6. Fractional combinatorial Calabi flow on surfaces

    Tianqi Wu and Xu Xu PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “combinatorial Calabi”, “Calabi flow”, “triangulated surfaces”

  7. Deformation of discrete conformal structures on surfaces

    Xu Xu PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “triangulated surfaces”, “combinatorial”, “vertex scalings”

  8. Combinatorial curvature flows for generalized hyperbolic circle packings

    Te Ba and Chao ZhengWider flows PDF

    Topics Discrete & graph Ricci, Other curvature flows, Kähler geometry · Shared terms “combinatorial Calabi”, “Calabi flow”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.