Paper · Apr 16, 2026

Integrable Deformations and Stability of the Ricci Flow

Maxwell Stolarski, Alex Waldron

math.DGarXiv:2604.15198
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Abstract

We provide a comparatively simple proof of the dynamical stability of Ricci flow near a linearly stable Ricci-flat ALE metric with integrable deformations. Our proof relies on the equivalence between integrability and an "almost-orthogonality" property of the Ricci-DeTurck tensor, allowing us to analyze the latter directly. We obtain our main results in weighted Holder spaces and then show how to recover the -stability theorems of Deruelle-Kroncke and Kroncke-Petersen.

Comments: 65 pages

Ricci flowRicci-flat & Calabi–YauStability & uniquenessHarmonic maps & DeTurck trick

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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 Convergence stability for Ricci flow on manifolds with bounded geometry (Eric Bahuaud, Christine Guenther, James Isenberg et al., 2023)1Bahuaud 2023 Ricci flow on Finsler manifolds (Behroz Bidabad and Maral K. Sedaghat, 2023)7Bidabad 2023 Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition (Man-Chun Lee and Stephen Shang Yi Liu, 2024)3Lee 2024 Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus (Pinsard Morel Lucas, 2025)5Lucas 2025 On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume (Ruojing Jiang and Franco Vargas Pallete, 2025)4Jiang 2025 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)8Kim 2025 Almost Hermitian geometry and generalized Ricci flow (Anna Fino and Julieth Saavedra, 2026)2Fino 2026 Pseudolocality and strong uniqueness for complete Ricci flows (Shaochuang Huang and Zhuo Peng, 2026)6Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al. PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “Hölder”

  2. Almost Hermitian geometry and generalized Ricci flow

    Anna Fino and Julieth Saavedra PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “integrable”

  3. Ricci-DeTurck Flow from Initial Metric with Morrey-type Integrability Condition

    Man-Chun Lee and Stephen Shang Yi Liu PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “integrability”

  4. On the stability of Ricci flow on hyperbolic 3-manifolds of finite volume

    Ruojing Jiang and Franco Vargas Pallete PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness · Shared terms “weighted Hölder”

  5. Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

    Pinsard Morel Lucas PDF

    Topics Harmonic maps & DeTurck trick, Ricci-flat & Calabi–Yau

  6. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness

  7. Ricci flow on Finsler manifolds

    Behroz Bidabad and Maral K. Sedaghat PDF

    Topics Harmonic maps & DeTurck trick, Stability & uniqueness

  8. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Stability & uniqueness

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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) 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) 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) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

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.