Paper · Jun 2, 2019

Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature

Guoqiang Wu, Shijin Zhang

math.DGarXiv:1906.00412
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As on arXiv

Abstract

In this short note, using an argument by Munteanu and Wang, we provide an alternative proof of the fact first obtained by Lei Ni that shrinking gradient Kähler-Ricci solitons with positive bisectional curvature must be compact.

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Submitted
Jun 2, 2019
Subjects
math.DG
Journal
Published in C. R. Math. Acad. Sci. Paris, 354(2016), No. 7, 713-716
Ricci solitonsKähler–Ricci solitonsKähler geometry

Its neighbourhood

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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 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 Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature (Shijin Zhang, 2019)1Zhang 2019 Uniqueness of shrinking gradient Kähler-Ricci solitons on non-compact toric manifolds (Charles Cifarelli, 2020)6Cifarelli 2020 A new complete two-dimensional shrinking gradient Kähler-Ricci soliton (Richard H. Bamler, Charles Cifarelli, Ronan J. Conlon et al., 2022)3Bamler 2022 Shrinking gradient Kähler-Ricci solitons are simply-connected (Carlos Esparza, 2025)4Esparza 2025 An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for t=0 (Ivin Babu, Ronan J. Conlon and Alix Deruelle, 2025)7Babu 2025 Uniqueness of shrinking Kähler-Ricci solitons on resolutions of Kähler cones (Ronan J. Conlon and Alix Deruelle, 2026)5Conlon 2026 L^p Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons (Guangwen Zhao, 2026)8Zhao 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)2Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Gradient Kähler-Ricci solitons with nonnegative orthogonal bisectional curvature

    Shijin Zhang

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

  2. 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”

  3. 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”

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

    Carlos Esparza

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

  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”, “shrinking gradient”

  6. 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”

  7. An Aubin continuity path for asymptotically conical toric shrinking gradient Kähler-Ricci solitons: openness and a solution for

    Ivin Babu, Ronan J. Conlon and Alix Deruelle

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

  8. Liouville theorems for pluriharmonic functions on gradient Kähler-Ricci solitons

    Guangwen Zhao

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

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.