Paper · Oct 27, 2022

A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

Huai-Dong Cao, Xiaofeng Sun, Yingying Zhang

math.DGarXiv:2210.15627J Geom Anal 32, 288 (2022)
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Abstract

In this paper we define a Weil-Petersson type metric on the space of shrinking Kaehler-Ricci solitons and prove a necessary and sufficient condition on when it is independent of the choices of Kaehler-Ricci soliton metrics. We also show that the Weil-Petersson metric is Kaehler when it defines a metric on the Kuranishi space of small deformations of Fano Kaehler-Ricci solitons. Finally, we establish the first and second order deformation of Fano Kähler-Ricci solitons and show that, essentially, the first effective term in deforming Kaehler-Ricci solitons leads to the Weil-Petersson metric.

Comments: 16 pages; final version available at the journal cite of JGEA (see the journal reference info below)

Ricci solitonsKähler–Ricci solitonsKähler geometry

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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 2021, 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 On canonical radial Kaehler metrics (Andrea Loi, Filippo Salis and Fabio Zuddas, 2021)1Loi 2021 Lower bound of modified K-energy on a Fano manifold with degeneration for Kähler-Ricci solitons (Liang Zhang, 2021)2Zhang 2021 Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (Huai-Dong Cao and Junming Xie, 2022)5Cao 2022 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)4Futaki 2023 L^p Liouville type theorems for harmonic functions on gradient Ricci solitons (Yong Luo, 2023)7Luo 2023 Generalized almost-Kähler-Ricci solitons (Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi, 2024)3Albanese 2024 Toric geometry of generalized Kähler-Ricci solitons (Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al., 2025)8Apostolov 2025 L^p Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons (Guangwen Zhao, 2026)6Zhao 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On canonical radial Kaehler metrics

    Andrea Loi, Filippo Salis and Fabio Zuddas PDF

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

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

    Liang Zhang PDF

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

  3. Generalized almost-Kähler-Ricci solitons

    Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi PDF

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

  4. Deformations of Fano manifolds with weighted solitons

    Akito Futaki PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Kaehler Ricci”, “Fano”, “small deformations”

  5. Four-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature

    Huai-Dong Cao and Junming Xie PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Huai-Dong Cao

  6. Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

    Guangwen Zhao PDF

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

  7. Liouville type theorems for harmonic functions on gradient Ricci solitons

    Yong Luo PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

  8. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al. PDF

    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. 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)

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.