Paper · Sep 30, 2021

On canonical radial Kaehler metrics

Andrea Loi, Filippo Salis, Fabio Zuddas

math.DGarXiv:2109.15229Osaka J. Math. 60 (3), 545-554 (july 2023)
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Abstract

We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g are constant; 3. g is extremal (not cscK) and one of its generalized scalar curvature is constant.

Comments: 11 pages. arXiv admin note: text overlap with arXiv:2105.10695

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryScalar curvatureKähler–Einstein & K-stability

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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 Holomorphic isometries into homogeneous bounded domains (Andrea Loi and Roberto Mossa, 2022)4Loi 2022 A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons (Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang, 2022)6Cao 2022 Deformations of Fano manifolds with weighted solitons (Akito Futaki, 2023)8Futaki 2023 Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds (A. Loi and R. Mossa, 2023)5Loi 2023 Ricci iterations of well-behaved Kähler metrics (Andrea Loi and Giovanni Placini, 2023)3Loi 2023 Universal embeddings of flag manifolds and rigidity phenomena (Andrea Loi, Roberto Mossa and Fabio Zuddas, 2025)2Loi 2025 On the Bergman metric of Cartan-Hartogs domains (Andrea Loi, Roberto Mossa and Fabio ZUddas, 2025)1Loi 2025 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)7Nitta 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 Bergman metric of Cartan-Hartogs domains

    Andrea Loi, Roberto Mossa and Fabio ZUddas PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same authors Andrea Loi, Fabio Zuddas

  2. Universal embeddings of flag manifolds and rigidity phenomena

    Andrea Loi, Roberto Mossa and Fabio Zuddas PDF

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Andrea Loi, Fabio Zuddas

  3. Ricci iterations of well-behaved Kähler metrics

    Andrea Loi and Giovanni Placini PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same author Andrea Loi

  4. Holomorphic isometries into homogeneous bounded domains

    Andrea Loi and Roberto Mossa PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same author Andrea Loi

  5. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same author Andrea Loi

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

    Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang PDF

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

  7. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta PDF

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

  8. Deformations of Fano manifolds with weighted solitons

    Akito Futaki PDF

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

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. 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.