Paper · Sep 18, 2026

Linear Stability of Steady and Expanding Kähler-Ricci Solitons

Lucas Lavoyer, Adam Thompson

math.DGarXiv:2609.21472
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Abstract

We prove linear stability of all steady and expanding gradient Kähler-Ricci solitons. In the expanding case, we prove strict linear stability under very general assumptions. In particular, every asymptotically conical expanding gradient Kähler-Ricci soliton is strictly linearly stable.

Comments: 14 pages, comments welcome

Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniquenessBoundaries & non-compact

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  1. A non-Kähler expanding Ricci soliton with a Kähler tangent cone at infinity

    Richard H. Bamler, Eric Chen and Ronan J. Conlon PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “expanding gradient”, “gradient Kähler”, “asymptotically conical”

  2. Linear stability and instability of Kähler Ricci solitons

    Keaton Naff and Tristan Ozuch PDF

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

  3. Stability of asymptotically conical gradient Kähler-Ricci expanders

    Longteng Chen PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “expanding Kähler”, “gradient Kähler”, “asymptotically conical”

  4. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “gradient Kähler”, “asymptotically conical”

  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 · Shared terms “gradient Kähler”

  6. An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for

    Ivin Babu, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “gradient Kähler”, “asymptotically conical”

  7. Uniqueness of asymptotically conical Kähler-Ricci flow

    Longteng Chen PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry · Shared terms “expanding Kähler”, “asymptotically conical”

  8. On geometry of steady toric Kähler-Ricci solitons

    Yury Ustinovskiy PDF

    Topics Kähler–Ricci solitons, Boundaries & non-compact, Kähler geometry

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 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) 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) 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.