Paper · Nov 19, 2025

Linear stability and instability of Kähler Ricci solitons

Keaton Naff, Tristan Ozuch

math.DGarXiv:2511.15885v2
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Abstract

We show that the recently discovered BCCD shrinking soliton is linearly unstable, by extending the approach of and, via recent work the on gradient shrinking Ricci solitons. On the other hand, we prove that the weighted -spectra of the weighted Lichnerowicz Laplacians of steady and expanding Kähler Ricci solitons are nonpositive in real dimension . We additionally determine the linear stability of the orbifold singularities of Kähler solitons: shrinkers are unstable, steadies are neutrally stable and expanders are strictly stable. All of these results follow from new Weitzenböck formulae for the weighted Lichnerowicz Laplacian specialized to Kähler metrics.

Comments: 20 pages; comments welcome! v2: Updated citations and added motivating questions

Ricci solitonsKähler–Ricci solitonsSingularitiesKähler geometryStability & uniquenessEigenvalues & spectrum

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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 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 Lower bound of modified K-energy on a Fano manifold with degeneration for Kähler-Ricci solitons (Liang Zhang, 2021)6Zhang 2021 Spinors and mass on weighted manifolds (Julius Baldauf and Tristan Ozuch, 2022)8Baldauf 2022 The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers (Fei He and Jianyu Ou, 2024)7He 2024 Orbifold singularity formation along ancient and immortal Ricci flows (Alix Deruelle and Tristan Ozuch, 2024)3Deruelle 2024 Linear stability of the blowdown Ricci shrinker in 4D (Keaton Naff and Tristan Ozuch, 2025)1Naff 2025 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)4Kim 2025 Linear Stability of Steady and Expanding Kähler-Ricci Solitons (Lucas Lavoyer and Adam Thompson, 2026)2Lavoyer 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)5Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Linear stability of the blowdown Ricci shrinker in 4D

    Keaton Naff and Tristan Ozuch PDF

    Topics Eigenvalues & spectrum, Kähler geometry, Singularities · Same authors Keaton Naff, Tristan Ozuch · Shared terms “weighted Lichnerowicz”, “linear stability”, “Lichnerowicz Laplacian”

  2. Linear Stability of Steady and Expanding Kähler-Ricci Solitons

    Lucas Lavoyer and Adam Thompson PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “linear stability”, “expanding Kähler”, “linearly”

  3. Orbifold singularity formation along ancient and immortal Ricci flows

    Alix Deruelle and Tristan Ozuch PDF

    Topics Singularities, Stability & uniqueness, Ricci solitons · Same author Tristan Ozuch · Shared terms “orbifold singularities”, “linear stability”, “real dimension”

  4. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Kähler geometry, Stability & uniqueness · Same author Tristan Ozuch · Shared terms “linear stability”, “unstable”

  5. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness

  6. Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons

    Liang Zhang PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness

  7. The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers

    Fei He and Jianyu Ou PDF

    Topics Eigenvalues & spectrum, Kähler geometry, Ricci solitons · Shared terms “Laplacian”, “gradient shrinking”, “shrinking Ricci”

  8. Spinors and mass on weighted manifolds

    Julius Baldauf and Tristan Ozuch PDF

    Topics Eigenvalues & spectrum · Same author Tristan Ozuch

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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 solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Kähler–Ricci solitons Kähler metrics whose Ricci form differs from the metric by a Lie derivative along a holomorphic vector field. Start with: Uniqueness of Kähler-Ricci solitons (Tian et al., 2000) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

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