Paper · Feb 12, 2024

A version of Bakry-Émery Ricci flow on a finite graph

Bobo Hua, Yong Lin, Tao Wang

math.DGarXiv:2402.07475
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Abstract

In this paper, we study the Bakry-Émery Ricci flow on finite graphs. Our main result is the local existence and uniqueness of solutions to the Ricci flow. We prove the long-time convergence or finite-time blow up for the Bakry-Émery Ricci flow on finite trees and circles.

Ricci flowSingularitiesLimit spaces & RCDStability & 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 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 Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition (Ling Wu, 2022)6Wu 2022 A combinatorial curvature flow in spherical background geometry (Huabin Ge, Bobo Hua and Puchun Zhou, 2023)7Ge 2023 On the Ricci flow on Trees (Shuliang Bai, Bobo Hua, Yong Lin et al., 2025)1Bai 2025 Rigidity of weighted manifolds via classification results for semilinear equations (Giulio Ciraolo, Alberto Farina and Troy Petitt, 2025)8Ciraolo 2025 The Ricci flow with prescribed curvature on graphs (Yong Lin and Shuang Liu, 2026)4Lin 2026 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)5Ye 2026 The Ollivier Ricci flow with prescribed curvature on infinite graphs (Bobo Hua, Yong Lin and Shuang Liu, 2026)2Hua 2026 Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees (Shuliang Bai and Bobo Hua, 2026)3Bai 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Ricci flow on Trees

    Shuliang Bai, Bobo Hua, Yong Lin et al. PDF

    Topics Discrete & graph Ricci · Same authors Bobo Hua, Yong Lin · Shared terms “finite trees”, “long time”

  2. The Ollivier Ricci flow with prescribed curvature on infinite graphs

    Bobo Hua, Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Stability & uniqueness · Same authors Bobo Hua, Yong Lin

  3. Curvature Diffusion of Inverse-weight Lin–Lu–Yau Ricci Flow on Finite Trees

    Shuliang Bai and Bobo Hua PDF

    Topics Discrete & graph Ricci · Same author Bobo Hua · Shared terms “finite trees”

  4. The Ricci flow with prescribed curvature on graphs

    Yong Lin and Shuang Liu PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence, Stability & uniqueness · Same author Yong Lin · Shared terms “finite graph”

  5. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu PDF

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”

  6. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu PDF

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”

  7. A combinatorial curvature flow in spherical background geometry

    Huabin Ge, Bobo Hua and Puchun Zhou PDF

    Topics Discrete & graph Ricci, Long-time existence & convergence · Same author Bobo Hua

  8. Rigidity of weighted manifolds via classification results for semilinear equations

    Giulio Ciraolo, Alberto Farina and Troy Petitt PDF

    Topics Limit spaces & RCD, Stability & uniqueness · Shared terms “Émery Ricci”, “Bakry Émery”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) 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)

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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.