Paper · Jul 5, 2021

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

Chi Li

math.DGarXiv:2107.02088
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Abstract

This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for -weighted Kähler-Ricci solitons (-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular -soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sasaki-Einstein metric to a particular -soliton equation on any quasi-regular quotient. We will revisit this transformation by understanding how the corresponding transversal complex Monge-Ampère equations are transformed under the deformation of Reeb vector fields. Finally we explain how this PDE/pluripotential point of view allows one to prove the YTD conjecture for general Fano cones by combining the YTD conjecture for -solitons on log Fano pairs and the above transformation.

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Submitted
Jul 5, 2021
Subjects
math.DG

Comments: 47 pages, comments welcome

Ricci solitonsKähler–Ricci solitonsEinstein metricsRicci-flat & Calabi–YauKähler geometryKähler–Einstein & K-stabilitySasakian & contact geometry

Its neighbourhood

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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 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 Coupled Sasaki-Ricci solitons (Akito Futaki and Yingying Zhang, 2018)6Futaki 2018 On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations (Jiyuan Han and Chi Li, 2020)1Han 2020 Irregular Eguchi-Hanson type metrics and their soliton analogues (Akito Futaki, 2020)8Futaki 2020 Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics (Vestislav Apostolov, Simon Jubert and Abdellah Lahdili, 2021)7Apostolov 2021 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)2Futaki 2023 Log del Pezzo ℂ^*-surfaces, Kähler-Einstein metrics, Kähler-Ricci solitons and Sasaki-Einstein metrics (Daniel Hättig, Jürgen Hausen and Hendrik Süß, 2023)4Hättig 2023 Kähler-Ricci shrinkers and Fano fibrations (Song Sun and Junsheng Zhang, 2024)5Sun 2024 On weighted extremal Kähler metrics (Akito Futaki, 2026)3Futaki 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci soliton equations

    Jiyuan Han and Chi Li

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Chi Li · Shared terms “YTD”, “Tian Donaldson”, “Yau Tian”

  2. Deformations of Fano manifolds with weighted solitons

    Akito Futaki

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Ricci-flat & Calabi–Yau · Shared terms “cone metrics”, “Sasaki Einstein”, “Fano”

  3. On weighted extremal Kähler metrics

    Akito Futaki

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

  4. 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”

  5. Kähler-Ricci shrinkers and Fano fibrations

    Song Sun and Junsheng Zhang

    Topics Kähler–Ricci solitons, Ricci-flat & Calabi–Yau, Einstein metrics · Shared terms “cone metrics”, “Kähler cone”, “YTD”

  6. Coupled Sasaki-Ricci solitons

    Akito Futaki and Yingying Zhang

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

  7. Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics

    Vestislav Apostolov, Simon Jubert and Abdellah Lahdili

    Topics Sasakian & contact geometry, Kähler–Ricci solitons, Ricci-flat & Calabi–Yau · Shared terms “Sasaki”, “Han”, “Fano”

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

    Akito Futaki

    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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… 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.