Paper · Aug 14, 2020

Irregular Eguchi-Hanson type metrics and their soliton analogues

Akito Futaki

math.DGarXiv:2008.07351v3
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Abstract

We verify the extension to the zero section of momentum construction of Kaehler-Einstein metrics and Kaehler-Ricci solitons on the total space Y of positive rational powers of the canonical line bundle of toric Fano manifolds with possibly irregular Sasaki-Einstein metrics. More precisely, we show that the extended metric along the zero section has an expression which can be extended to Y, restricts to the associated unit circle bundle as a transversely Kaehler-Einstein (Sasakian eta-Einstein) metric scaled in the Reeb flow direction, and that there is a Riemannian submersion from the scaled Sasakian eta-Einstein metric to the induced metric of the zero section.

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Submitted
Aug 14, 2020
Last revised
Feb 1, 2021 (v3)
Subjects
math.DG

Comments: Title changed. Abstract changed. Text substantially revised. References added

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry

Its neighbourhood

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 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 Volume minimization and Conformally Kähler, Einstein-Maxwell geometry (Akito Futaki and Hajime Ono, 2017)6Futaki 2017 Toric extremal Kaehler-Ricci solitons are Kaehler-Einstein (Simone Calamai and David Petrecca, 2017)5Calamai 2017 Volume minimization and obstructions to solving some problems in Kähler geometry (Akito Futaki and Hajime Ono, 2018)3Futaki 2018 Coupled Sasaki-Ricci solitons (Akito Futaki and Yingying Zhang, 2018)2Futaki 2018 Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics (Chi Li, 2021)8Li 2021 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)1Futaki 2023 Log del Pezzo ℂ^*-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics (Daniel Hättig, Jürgen Hausen and Hendrik Süß, 2023)7Hättig 2023 On weighted extremal Kähler metrics (Akito Futaki, 2026)4Futaki 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Same author Akito Futaki · Shared terms “zero section”, “Kaehler Einstein”, “Fano manifolds”

  2. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Same author Akito Futaki · Shared terms “Sasaki Einstein”, “extension”

  3. Volume minimization and obstructions to solving some problems in Kähler geometry

    Akito Futaki and Hajime Ono

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Same author Akito Futaki · Shared terms “extended”, “Sasaki Einstein”

  4. On weighted extremal Kähler metrics

    Akito Futaki

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Same author Akito Futaki · Shared terms “Sasaki Einstein”

  5. Toric extremal Kaehler-Ricci solitons are Kaehler-Einstein

    Simone Calamai and David Petrecca

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Shared terms “Kaehler Einstein”, “Kaehler Ricci”

  6. Volume minimization and Conformally Kähler, Einstein-Maxwell geometry

    Akito Futaki and Hajime Ono

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Einstein metrics · Same author Akito Futaki · Shared terms “Sasaki Einstein”

  7. Log del Pezzo -surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics

    Daniel Hättig, Jürgen Hausen and Hendrik Süß

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Shared terms “Sasaki Einstein”

  8. Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics

    Chi Li

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Shared terms “irregular”, “Sasaki Einstein”, “Reeb”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.