Paper · Mar 2, 2020

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

Andrea Loi, Roberto Mossa

math.DGarXiv:2003.00841v2
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Abstract

Let be a Kähler-Ricci soliton on a complex manifold . We prove that if the Kähler manifold can be Kähler immersed into a definite or indefinite complex space form of constant holomorphic sectional curvature , then is Einstein. Moreover, its Einstein constant is a rational multiple of .

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Submitted
Mar 2, 2020
Last revised
Apr 20, 2021 (v2)
Subjects
math.DG, math.CV
Journal
Proc. Amer. Math. Soc. 149 (2021), 4931-4941
DOI
10.1090/proc/15628

Comments: 12 pages. To appear in Proc. Amer. Math. Soc

Ricci solitonsKähler–Ricci solitonsSphere theoremsKähler geometry

Its neighbourhood

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  1. Kaehler Ricci solitons induced by infinite dimensional complex space forms

    Andrea Loi, Fabio Zuddas and Filippo Salis

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

  2. Holomorphic isometries into homogeneous bounded domains

    Andrea Loi and Roberto Mossa

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Andrea Loi, Roberto Mossa · Shared terms “indefinite”, “holomorphic”

  3. 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, Roberto Mossa · Shared terms “indefinite”, “holomorphic”

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

    Roberto Mossa and Giovanni Placini

    Topics Sphere theorems, Kähler–Ricci solitons, Kähler geometry · Same author Roberto Mossa · Shared terms “immersion”, “space form”, “holomorphic”

  5. Ricci iterations of well-behaved Kähler metrics

    Andrea Loi and Giovanni Placini

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

  6. Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

    A. Loi and R. Mossa

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Andrea Loi, Roberto Mossa · Shared terms “holomorphic”

  7. On the Bergman metric of Cartan-Hartogs domains

    Andrea Loi, Roberto Mossa and Fabio ZUddas

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same authors Andrea Loi, Roberto Mossa

  8. Real hypersurfaces with -Ricci solitons of non-flat complex space forms

    Xiaomin Chen

    Topics Sphere theorems, Ricci solitons · Shared terms “complex space”, “space form”, “holomorphic”

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) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… 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.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.