Paper · Oct 1, 2026

Generalized scalar curvature and Ricci flow on compact Lie groups

Zhiqi Chen, Jianhua Yu, Huibin Chen, Junna Ni

math.DGarXiv:2610.03795
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Abstract

Let be a compact connected semisimple Lie group, let be a bi-invariant metric and let be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauScalar curvatureHomogeneous spacesLong-time existence & convergence

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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 Bismut Ricci flat manifolds with symmetries (Fabio Podestà and Alberto Raffero, 2022)5Podestà 2022 Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum) (Jorge Lauret and Cynthia E. Will, 2023)6Lauret 2023 Generalized Ricci flow on aligned homogeneous spaces (Valeria Gutiérrez, 2024)4Gutiérrez 2024 The homogeneous generalized Ricci flow (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2024)8Fusi 2024 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)7Giudice 2025 Homogeneous Generalized Ricci flows II (Elia Fusi, Ramiro A. Lafuente and James Stanfield, 2026)1Fusi 2026 Harmonicity and Existence of Algebraic Generalized Ricci Solitons (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)2Chen 2026 Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions (Huibin Chen, Zhiqi Chen and Fuhai Zhu, 2026)3Chen 2026
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  1. Homogeneous Generalized Ricci flows II

    Elia Fusi, Ramiro A. Lafuente and James Stanfield PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Lie groups”, “Bismut”, “left invariant”

  2. Harmonicity and Existence of Algebraic Generalized Ricci Solitons

    Huibin Chen, Zhiqi Chen and Fuhai Zhu PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Same authors Zhiqi Chen, Huibin Chen · Shared terms “Lie algebras”, “nilpotent Lie”, “Killing”

  3. Harmonic torsion of algebraic generalized Ricci solitons and the structure of solvable extensions

    Huibin Chen, Zhiqi Chen and Fuhai Zhu PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Same authors Zhiqi Chen, Huibin Chen · Shared terms “Lie algebras”

  4. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut Ricci”, “compact simple”, “Lie groups”

  5. Bismut Ricci flat manifolds with symmetries

    Fabio Podestà and Alberto Raffero PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut Ricci”, “invariant Bismut”

  6. Bismut Ricci flat generalized metrics on compact homogeneous spaces (including a Corrigendum)

    Jorge Lauret and Cynthia E. Will PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Homogeneous spaces · Shared terms “Bismut Ricci”, “invariant generalized”, “semisimple”

  7. Positive Hermitian curvature flow on 2-step nilpotent Lie groups

    Ettore Lo Giudice PDF

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “Lie groups”, “nilpotent Lie”, “left invariant”

  8. The homogeneous generalized Ricci flow

    Elia Fusi, Ramiro A. Lafuente and James Stanfield PDF

    Topics Generalized & G₂ flows, Homogeneous spaces, Long-time existence & convergence · Shared terms “Lie groups”, “invariant generalized”, “Lauret”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 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.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.