Paper · May 9, 2023

Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

A. Loi, R. Mossa

math.DGarXiv:2305.05244v3Proc. Amer. Math. Soc.152 (2024), no.7, 3051-3062
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Abstract

We prove two rigidity results on holomorphic isometries into homogeneous Kähler manifolds. The first shows that a Kähler-Ricci soliton induced by the homogeneous metric of the Kähler product of a special flag manifold (i.e. a flag of classical type or integral type) with a bounded homogeneous domain is trivial, i.e. Kähler-Einstein. In the second one we prove that: (i) a flat space is not relative to the Kähler product of a special flag manifold with a homogeneous bounded domain, (ii) a special flag manifold is not relative to the Kähler product of a flat space with a homogeneous bounded domain and (iii) a homogeneous bounded domain is not relative to the Kähler product of a flat space with a special flag manifold. Our theorems strongly extend the results in [4], [5], [12], [13] and [22].

Comments: 14 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler geometryHomogeneous spacesStability & uniquenessKähler–Einstein & K-stability

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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 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)6Loi 2021 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)8Nakagawa 2022 Holomorphic isometries into homogeneous bounded domains (Andrea Loi and Roberto Mossa, 2022)1Loi 2022 Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds (Roberto Mossa and Giovanni Placini, 2023)4Mossa 2023 Ricci iterations of well-behaved Kähler metrics (Andrea Loi and Giovanni Placini, 2023)5Loi 2023 The Kähler-Ricci soliton on bounded pseudoconvex domains (Zehao Sha, 2024)7Sha 2024 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)3Loi 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Holomorphic isometries into homogeneous bounded domains

    Andrea Loi and Roberto Mossa PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics · Same authors A. Loi, R. Mossa · Shared terms “homogeneous bounded”, “holomorphic isometries”, “bounded domain”

  2. Universal embeddings of flag manifolds and rigidity phenomena

    Andrea Loi, Roberto Mossa and Fabio Zuddas PDF

    Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same authors A. Loi, R. Mossa · Shared terms “flag manifold”, “homogeneous bounded”, “bounded domain”

  3. 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 A. Loi, R. Mossa · Shared terms “bounded domain”, “III”

  4. Immersions of Sasaki-Ricci solitons into homogeneous Sasakian manifolds

    Roberto Mossa and Giovanni Placini PDF

    Topics Kähler–Ricci solitons, Homogeneous spaces, Kähler geometry · Same author R. Mossa · Shared terms “holomorphic isometries”, “homogeneous bounded”, “flag manifold”

  5. 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 A. Loi

  6. On canonical radial Kaehler metrics

    Andrea Loi, Filippo Salis and Fabio Zuddas PDF

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

  7. The Kähler-Ricci soliton on bounded pseudoconvex domains

    Zehao Sha PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

  8. Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

    Yasuhiro Nakagawa and Satoshi Nakamura PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Einstein metrics

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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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  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.