Paper · Jul 16, 2025

The existence and uniqueness of infinite combinatorial Yamabe flows

Bohao Ji

math.DGWider flowsarXiv:2507.12355
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Abstract

In this paper, we study the combinatorial Yamabe flow on infinite triangulated surfaces in Euclidean background geometry, aiming for solving discrete Yamabe problem on noncompact surfaces. Under suitable conditions, we establish the short-time existence and uniqueness of the flow. We further introduce an extended version of the flow and prove its long-time existence. As an application, we prove the convergence result of the Yamabe flow in the case of hexagonal triangulations of the plane.

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Submitted
Jul 16, 2025
Subjects
math.DG
Yamabe flowStability & uniquenessBoundaries & non-compactLong-time existence & convergenceHarmonic maps & DeTurck trickDiscrete & 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 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 Combinatorial Calabi flows on surfaces with boundary (Yanwen Luo and Xu Xu, 2021)3Luo 2021 Infinite-time incompleteness of noncompact Yamabe flow (Jin Takahashi and Hikaru Yamamoto, 2021)4Takahashi 2021 Combinatorial Yamabe flow on hyperbolic bordered surfaces (Shengyu Li, Xu Xu and Ze Zhou, 2022)5Li 2022 Existence of the (α,β)-Ricci-Yamabe flow on closed manifolds (Liangdi Zhang, 2023)6Zhang 2023 Generalized Yamabe Flows (Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro Vitti, 2024)7Lye 2024 Infinite Combinatorial Yamabe Flows in Three Dimensions (Bohao Ji, 2026)1Ji 2026 Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces (Yuerong Bian and Xu Xu, 2026)2Bian 2026 Combinatorial Calabi flows on infinitely triangulated surfaces (Yuerong Bian and Xu Xu, 2026)8Bian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinite Combinatorial Yamabe Flows in Three Dimensions

    Bohao JiWider flows

    Topics Yamabe flow, Harmonic maps & DeTurck trick, Discrete & graph Ricci · Same author Bohao Ji · Shared terms “infinite combinatorial”, “combinatorial Yamabe”, “background geometry”

  2. Combinatorial Yamabe flow on infinitely triangulated hyperbolic surfaces

    Yuerong Bian and Xu XuWider flows

    Topics Yamabe flow, Harmonic maps & DeTurck trick, Discrete & graph Ricci · Shared terms “combinatorial Yamabe”, “triangulated surfaces”, “extended”

  3. Combinatorial Calabi flows on surfaces with boundary

    Yanwen Luo and Xu XuWider flows

    Topics Yamabe flow, Discrete & graph Ricci, Long-time existence & convergence · Shared terms “combinatorial Yamabe”, “triangulated surfaces”

  4. Infinite-time incompleteness of noncompact Yamabe flow

    Jin Takahashi and Hikaru YamamotoWider flows

    Topics Yamabe flow, Long-time existence & convergence, Boundaries & non-compact

  5. Combinatorial Yamabe flow on hyperbolic bordered surfaces

    Shengyu Li, Xu Xu and Ze ZhouWider flows

    Topics Yamabe flow, Long-time existence & convergence · Shared terms “combinatorial Yamabe”

  6. Existence of the -Ricci-Yamabe flow on closed manifolds

    Liangdi Zhang

    Topics Yamabe flow, Harmonic maps & DeTurck trick, Long-time existence & convergence · Shared terms “short time”

  7. Generalized Yamabe Flows

    Jørgen Olsen Lye, Boris Vertman and Mannaim Gennaro VittiWider flows

    Topics Yamabe flow, Long-time existence & convergence

  8. Combinatorial Calabi flows on infinitely triangulated surfaces

    Yuerong Bian and Xu XuWider flows

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

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.

Yamabe flow Wider flows The Yamabe flow deforms a metric within its conformal class, driven by scalar curvature. It is a parabolic approach to the Yamabe problem: find a… Start with: The yamabe flow on locally conformally flat manifolds with positive ricci curvature (Chow, 1992) 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) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 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.