Paper · Jun 2, 2019

Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature

Shijin Zhang

math.DGarXiv:1906.00415
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Abstract

In this paper, we prove that any complete shrinking gradient Kähler-Ricci solitons with positive orthogonal bisectional curvature must be compact. We also obtain a classification of the complete shrinking gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature.

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Submitted
Jun 2, 2019
Subjects
math.DG
DOI
10.1007/s00025-019-1052-5

Comments: to appear in Results in Math.. Xiaolong Li and Lei Ni obtained the same result in their paper "Khler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature" (see arXiv:1903.02615). Then even obtained more interesting results. arXiv admin note: text overlap with arXiv:1906.00412

Ricci solitonsKähler–Ricci solitonsKähler geometry

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 Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature (Xiaolong Li and Lei Ni, 2019)2Li 2019 Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature (Guoqiang Wu and Shijin Zhang, 2019)1Wu 2019 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)6Lee 2019 Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds (Charles Cifarelli, 2020)7Cifarelli 2020 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)4Bamler 2022 Shrinking gradient Kähler-Ricci solitons are simply-connected (Carlos Esparza, 2025)8Esparza 2025 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)5Conlon 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)3Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature

    Guoqiang Wu and Shijin Zhang

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Shijin Zhang · Shared terms “bisectional curvature”, “gradient Kähler”, “shrinking gradient”

  2. Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature

    Xiaolong Li and Lei Ni

    Topics Kähler geometry, Ricci solitons · Shared terms “orthogonal bisectional”, “nonnegative orthogonal”, “bisectional curvature”

  3. Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “gradient Kähler”, “shrinking gradient”, “complete shrinking”

  4. A new complete two-dimensional shrinking gradient Kähler-Ricci soliton

    Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al.

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “gradient Kähler”, “shrinking gradient”, “complete shrinking”

  5. Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones

    Ronan J. Conlon and Alix Deruelle

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “gradient Kähler”, “complete shrinking”, “shrinking gradient”

  6. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

    Topics Kähler geometry · Shared terms “orthogonal bisectional”, “bisectional curvature”, “nonnegative orthogonal”

  7. Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds

    Charles Cifarelli

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “gradient Kähler”, “shrinking gradient”, “complete shrinking”

  8. Shrinking gradient Kähler-Ricci solitons are simply-connected

    Carlos Esparza

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

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 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. 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.