Paper · May 22, 2021

Kaehler Ricci solitons induced by infinite dimensional complex space forms

Andrea Loi, Fabio Zuddas, Filippo Salis

math.DGarXiv:2105.10695
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Abstract

We exhibit families of non trivial (i.e. not Kaehler-Einstein) radial Kaehler-Ricci solitons (KRS), both complete and not complete, which can be Kaehler immersed into infinite dimensional complex space forms. This result shows that the triviality of a KRS induced by a finite dimensional complex space form proved in [12] does not hold when the ambient space is allowed to be infinite dimensional. Moreover, we show that the radial potential of a radial KRS induced by a non-elliptic complex space form is necessarily defined at the origin.

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Submitted
May 22, 2021
Subjects
math.DG, math.CV
Journal
Pacific J. Math. 316 (2022) 183-205
DOI
10.2140/pjm.2022.316.183

Comments: 22 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsSphere theoremsKähler geometryKähler–Einstein & K-stability

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 Toric extremal Kaehler-Ricci solitons are Kaehler-Einstein (Simone Calamai and David Petrecca, 2017)5Calamai 2017 Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space forms (Andrea Loi and Roberto Mossa, 2020)2Loi 2020 On canonical radial Kaehler metrics (Andrea Loi, Filippo Salis and Fabio Zuddas, 2021)1Loi 2021 Holomorphic isometries into homogeneous bounded domains (Andrea Loi and Roberto Mossa, 2022)7Loi 2022 Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds (A. Loi and R. Mossa, 2023)8Loi 2023 Ricci iterations of well-behaved Kähler metrics (Andrea Loi and Giovanni Placini, 2023)4Loi 2023 Universal embeddings of flag manifolds and rigidity phenomena (Andrea Loi, Roberto Mossa and Fabio Zuddas, 2025)6Loi 2025 On the Bergman metric of Cartan-Hartogs domains (Andrea Loi, Roberto Mossa and Fabio ZUddas, 2025)3Loi 2025
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

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same authors Andrea Loi, Fabio Zuddas, Filippo Salis · Shared terms “Kaehler”

  2. Kähler immersions of Kähler-Ricci solitons into definite or indefinite complex space forms

    Andrea Loi and Roberto Mossa

    Topics Sphere theorems, Kähler–Ricci solitons, Kähler geometry · Same author Andrea Loi · Shared terms “complex space”, “space form”

  3. On the Bergman metric of Cartan-Hartogs domains

    Andrea Loi, Roberto Mossa and Fabio ZUddas

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

  4. Ricci iterations of well-behaved Kähler metrics

    Andrea Loi and Giovanni Placini

    Topics Sphere theorems, Kähler–Ricci solitons, Kähler–Einstein & K-stability · Same author Andrea Loi · Shared terms “complex space”, “space form”

  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 Ricci”, “Kaehler Einstein”

  6. Universal embeddings of flag manifolds and rigidity phenomena

    Andrea Loi, Roberto Mossa and Fabio Zuddas

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Andrea Loi, Fabio Zuddas · Shared terms “triviality”

  7. Holomorphic isometries into homogeneous bounded domains

    Andrea Loi and Roberto Mossa

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Andrea Loi · Shared terms “soliton induced”

  8. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa

    Topics Kähler–Ricci solitons, Kähler–Einstein & K-stability, Einstein metrics · Same author Andrea Loi · Shared terms “soliton induced”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

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

  3. 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.