Paper · Sep 4, 2017

Dynamic instability of under Ricci flow

Dan Knopf, Natasa Sesum

math.DGarXiv:1709.01005v2
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Abstract

The intent of this short note is to provide context for and an independent proof of the discovery of Klaus Kroencke that complex projective space with its canonical Fubini–Study metric is dynamically unstable under Ricci flow in all complex dimensions N>1. The unstable perturbation is not Kaehler. This provides a counterexample to a well known conjecture widely attributed to Hamilton. Moreover, it shows that the expected stability of the subspace of Kaehler metrics under Ricci flow, another conjecture believed by several experts, needs to be interpreted in a more nuanced way than some may have expected.

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Submitted
Sep 4, 2017
Last revised
Apr 6, 2018 (v2)
Subjects
math.DG

Comments: Corrected an error in what had been Lemma 5. Revised version to appear in The Journal of Geometric Analysis

Ricci flowKähler geometryStability & uniqueness

Its neighbourhood

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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 2016, 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 Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons (James Isenberg, Dan Knopf and Natasa Sesum, 2017)1Isenberg 2017 The canonical Einstein metric on G₂ is dynamically unstable under the Ricci flow (Stuart James Hall, 2018)6Hall 2018 On the linear stability of nearly-Kähler 6-manifolds (Changliang Wang and M. Y. -K. Wang, 2019)7Wang 2019 Ricci Solitons, Conical Singularities, and Nonuniqueness (Sigurd B. Angenent and Dan Knopf, 2019)8Angenent 2019 Singularity formation of complete Ricci flow solutions (Timothy Carson, James Isenberg, Dan Knopf et al., 2020)3Carson 2020 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)2Garfinkle 2024 Local singularities of compact multiply warped Ricci flow solutions (James Isenberg, Dan Knopf, Zilu Ma et al., 2025)5Isenberg 2025 Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions (Sigurd B. Angenent and Dan Knopf, 2025)4Angenent 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Non-Kahler Ricci flow singularities modeled on Kahler-Ricci solitons

    James Isenberg, Dan Knopf and Natasa Sesum

    Topics Kähler geometry, Stability & uniqueness · Same authors Dan Knopf, Natasa Sesum · Shared terms “expected”, “subspace”

  2. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Stability & uniqueness · Same author Dan Knopf · Shared terms “unstable perturbations”, “Fubini”

  3. Singularity formation of complete Ricci flow solutions

    Timothy Carson, James Isenberg, Dan Knopf et al.

    Topics Stability & uniqueness · Same authors Dan Knopf, Natasa Sesum

  4. Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

    Sigurd B. Angenent and Dan Knopf

    Topics Stability & uniqueness · Same author Dan Knopf · Shared terms “instability”, “unstable perturbations”

  5. Local singularities of compact multiply warped Ricci flow solutions

    James Isenberg, Dan Knopf, Zilu Ma et al.

    Topics Ricci flow · Same authors Dan Knopf, Natasa Sesum

  6. The canonical Einstein metric on is dynamically unstable under the Ricci flow

    Stuart James Hall

    Topics Stability & uniqueness · Shared terms “dynamically unstable”

  7. On the linear stability of nearly-Kähler -manifolds

    Changliang Wang and M. Y. -K. Wang

    Topics Kähler geometry, Stability & uniqueness · Shared terms “unstable”, “dynamically”

  8. Ricci Solitons, Conical Singularities, and Nonuniqueness

    Sigurd B. Angenent and Dan Knopf

    Topics Stability & uniqueness · Same author Dan Knopf · Shared terms “expected”

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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… 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)

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.