Paper · Apr 20, 2026

The perturbative Ricci flow in gravity

Robert V. Harlander, Yannick Kluth, Jonas T. Kohnen, Henry Werthenbach

hep-tharXiv:2604.18678
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As on arXiv

Abstract

We develop a perturbative formulation of the Ricci flow in gravity. Following steps analogous to the gradient flow in QCD, we supplement the usual Feynman rules for perturbative gravity by flowed propagators and vertices as well as graviton flow lines which describe the evolution of gravity along the Ricci flow. By calculating vacuum expectation values of a number of independent operators at the two-loop level, we derive the required counterterms of the flowed action. Our results allow us to define a Ricci-flow based renormalization scheme for Newton's constant . Studying its renormalization group behavior, we recover a non-Gaußian fixed point in accordance with well-known non-perturbative considerations

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Submitted
Apr 20, 2026
Subjects
hep-th, gr-qc, hep-lat

Comments: 5 pages, 2 figures

Ricci flowPhysics

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 Stochastic Quantization of General Relativity à la Ricci-Flow (Matteo Lulli, Antonino Marciano and Xiaowen Shan, 2021)8Lulli 2021 Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity (A. A. Varshovi, 2022)2Varshovi 2022 Single copy of the Ricci flow (Rashid Alawadhi, 2022)3Alawadhi 2022 General Relativity and the Ricci Flow (Mohammed Alzain, 2022)7Alzain 2022 Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime (M. J. Luo, 2022)6Luo 2022 A path integral formula of quantum gravity emergent from entangled local structures (Jinglong Liu, Stephon Alexander, Antonino Marciano et al., 2023)1Liu 2023 Monotonicity of RG flow in emergent dual holography of worldsheet nonlinear σ model (Ki-Seok Kim, Arpita Mitra, Debangshu Mukherjee et al., 2024)5Kim 2024 Topologically Stabilized Torsion in Weak-Field Gravity: A Ricci-Flow Framework (Elisa Varani, 2025)4Varani 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A path integral formula of quantum gravity emergent from entangled local structures

    Jinglong Liu, Stephon Alexander, Antonino Marciano et al.

    Topics Physics · Shared terms “renormalization group”, “formulation”

  2. Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman Path-Integral Measure for Quantum Fields in the Presence of Gravity

    A. A. Varshovi

    Topics Physics · Shared terms “Feynman”, “formulation”, “studying”

  3. Single copy of the Ricci flow

    Rashid Alawadhi

    Topics Physics · Shared terms “perturbative”

  4. Topologically Stabilized Torsion in Weak-Field Gravity: A Ricci-Flow Framework

    Elisa Varani

    Topics Physics · Shared terms “fixed point”, “formulation”, “independent”

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

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

    Topics Physics · Shared terms “perturbative”, “renormalization group”, “analogous”

  6. Quantum Modified Gravity at Low Energy in the Ricci Flow of Quantum Spacetime

    M. J. Luo

    Topics Physics · Shared terms “renormalization”

  7. General Relativity and the Ricci Flow

    Mohammed Alzain

    Topics Physics

  8. Stochastic Quantization of General Relativity à la Ricci-Flow

    Matteo Lulli, Antonino Marciano and Xiaowen Shan

    Topics Physics · Shared terms “renormalization group”

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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.