Paper · Dec 10, 2018

Coupled Sasaki-Ricci solitons

Akito Futaki, Yingying Zhang

math.DGarXiv:1812.03784v2
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Abstract

Motivated by the study of coupled Kähler-Einstein metrics by Hultgren and Witt Nyström and coupled Kähler-Ricci solitons by Hultgren, we study in this paper coupled Sasaki-Einstein metrics and coupled Sasaki-Ricci solitons. We first show an isomorphism between the Lie algebra of all transverse holomorphic vector fields and certain space of coupled basic functions related to coupled twisted Laplacians for basic functions, and obtain extensions of the well-known obstructions to the existence of Kähler-Einstein metrics to this coupled case. These results are reduced to coupled Kähler-Einstein metrics when the Sasaki structure is regular. Secondly we show the existence of toric coupled Sasaki-Einstein metrics when the basic first Chern class is positive extending the work of Hultgren.

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Submitted
Dec 10, 2018
Last revised
Feb 25, 2019 (v2)
Subjects
math.DG
DOI
10.1007/s11425-018-9499-y

Comments: Final version. To appear in Science China Math

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)4Futaki 2017 Coupled Kähler-Ricci solitons on toric Fano manifolds (Jakob Hultgren, 2017)8Hultgren 2017 Volume minimization and obstructions to solving some problems in Kähler geometry (Akito Futaki and Hajime Ono, 2018)1Futaki 2018 Transverse Kähler-Ricci flow and deformations of the metric on the Sasaki space T^(1,1) (Vladimir Slesar, Mihai Visinescu and Gabriel-Eduard Vîlcu, 2019)7Slesar 2019 Irregular Eguchi-Hanson type metrics and their soliton analogues (Akito Futaki, 2020)3Futaki 2020 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)5Futaki 2023 Log del Pezzo ℂ^*-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics (Daniel Hättig, Jürgen Hausen and Hendrik Süß, 2023)6Hättig 2023 On weighted extremal Kähler metrics (Akito Futaki, 2026)2Futaki 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 “Sasaki Einstein”, “obstruction”, “vector field”

  2. 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”

  3. Irregular Eguchi-Hanson type metrics and their soliton analogues

    Akito Futaki

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

  4. 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”, “obstruction”, “vector field”

  5. 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 “Sasaki Einstein”

  6. 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”

  7. Transverse Kähler-Ricci flow and deformations of the metric on the Sasaki space

    Vladimir Slesar, Mihai Visinescu and Gabriel-Eduard Vîlcu

    Topics Sasakian & contact geometry, Einstein metrics, Kähler geometry · Shared terms “Sasaki”, “basic functions”, “transverse”

  8. Coupled Kähler-Ricci solitons on toric Fano manifolds

    Jakob Hultgren

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

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.