Paper · Jan 27, 2020

Existence and uniqueness of -invariant Kähler-Ricci solitons

Johannes Schäfer

math.DGarXiv:2001.09858v3
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Abstract

We use the momentum construction for -invariant Kähler metrics as developed by Hwang-Singer to construct new examples of steady Kähler-Ricci solitons. We also prove that these solitons are unique in their Kähler class, provided the vector field and the asymptotic behaviour are fixed.

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Submitted
Jan 27, 2020
Last revised
Aug 15, 2021 (v3)
Subjects
math.DG

Comments: 36 pages; results unchanged, but corrected proof of Lemma 4.4, removed Lemma 5.8 and added details to Section 5; to be published in the Annales de la Faculté des Sciences de Toulouse

Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness

Its neighbourhood

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  1. Asymptotically cylindrical steady Kähler-Ricci solitons

    Johannes Schäfer

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Johannes Schäfer · Shared terms “steady Kähler”

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

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “vector field”

  3. Steady Kähler-Ricci solitons on crepant resolutions of finite quotients of

    Olivier Biquard and Heather Macbeth

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “steady Kähler”

  4. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

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

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

    Liang Zhang

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

  6. On geometry of steady toric Kähler-Ricci solitons

    Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Shared terms “steady Kähler”

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

    Ronan J. Conlon and Alix Deruelle

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

  8. Kähler-Ricci solitons on horospherical variety

    François Delgove

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

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 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. 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.