Paper · Sep 23, 2026

On the singularity formation of gauge fields coupled to Ricci flow

Andoni Royo Abrego

math.DGarXiv:2609.27644
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Abstract

We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a bundle over .

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Submitted
Sep 23, 2026
Subjects
math.DG

Comments: Comments are welcome!

Ricci flowSingularitiesPerelman entropy

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 local Sobolev inequality on Ricci flow and its applications (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2021)5Chan 2021 Geometric flows and the Swampland (Davide De Biasio, 2023)8De Biasio 2023 Local smooth convergence of F-limit flows (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2023)6Chan 2023 Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear σ model (Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al., 2024)4Kim 2024 Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds (Zeraoulia Rafik and Sobhan Sobhan Allah, 2024)2Rafik 2024 The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory (Sergiu I. Vacaru and Elşen V. Veliev, 2025)7Vacaru 2025 Stochastic Thermodynamics on Time-Evolving Curved Spaces (Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al., 2026)3Nagase 2026 On the geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness (Yu Li, Longhuan Yan and Wenjia Zhang, 2026)1Li 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 geometry of asymptotically conical Ricci shrinkers (II): strong uniqueness

    Yu Li, Longhuan Yan and Wenjia Zhang

    Topics Perelman entropy, Singularities

  2. Innovative Dynamics: Utilizing Perelman's Entropy and Ricci Flow for Settler Position Models on Manifolds

    Zeraoulia Rafik and Sobhan Sobhan Allah

    Topics Perelman entropy

  3. Stochastic Thermodynamics on Time-Evolving Curved Spaces

    Rihito Nagase, Shoki Sugimoto, Asuka Takatsu et al.

    Topics Perelman entropy

  4. Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear model

    Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al.

    Topics Perelman entropy

  5. A local Sobolev inequality on Ricci flow and its applications

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

  6. Local smooth convergence of -limit flows

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang

    Topics Perelman entropy

  7. The fundamental physical importance of generic off-diagonal solutions and Grigori Perelman entropy in the Einstein gravity theory

    Sergiu I. Vacaru and Elşen V. Veliev

    Topics Perelman entropy

  8. Geometric flows and the Swampland

    Davide De Biasio

    Topics Perelman entropy

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) Perelman entropy Perelman found that Ricci flow is a gradient flow. His 𝓕-functional and 𝓦-entropy increase along the flow, and their monotonicity rules out… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.