Paper · Dec 17, 2017

Ricci solitons

Andrzej Derdzinski

math.DGarXiv:1712.06055
View PDF View HTML arXiv page

As on arXiv

Abstract

English translation of "Solitony Ricciego" (Wiadomości Matematyczne 48, 2012, no. 1, pp. 1-32). Despite the general-sounding title, the text covers just a few narrow topics: Perelman's proof of the fact that compact Ricci solitons are of the gradient type, and a detailed unified description of Page's and Berard Bergery's Einstein manifolds on the one hand, and Koiso's and Cao's Kähler-Ricci solitons on the other, as the Ricci solitons arising via conformal changes from certain special Kähler manifolds.

View PDF View HTML arXiv page
Submitted
Dec 17, 2017
Subjects
math.DG

Comments: 32 pages

Ricci solitonsKähler–Ricci solitonsEinstein metricsKähler 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 Concavity of Perelman's 𝓦-functional over the space of Kähler potentials (Nefton Pali, 2016)4Pali 2016 Kähler-Ricci solitons on horospherical variety (François Delgove, 2017)7Delgove 2017 Remarks on shrinking gradient Kähler-Ricci solitons with positive bisectional curvature (Guoqiang Wu and Shijin Zhang, 2019)8Wu 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)5Guo 2020 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)6Nakagawa 2022 Special Ricci-Hessian equations on Kähler manifolds (Andrzej Derdzinski and Paolo Piccione, 2023)1Derdzinski 2023 The Kähler-Ricci soliton on bounded pseudoconvex domains (Zehao Sha, 2024)3Sha 2024 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)2Nitta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Special Ricci-Hessian equations on Kähler manifolds

    Andrzej Derdzinski and Paolo Piccione

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Same author Andrzej Derdzinski · Shared terms “special Kähler”

  2. Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein

    Yasufumi Nitta

    Topics Kähler–Ricci solitons, Einstein metrics, Kähler geometry

  3. The Kähler-Ricci soliton on bounded pseudoconvex domains

    Zehao Sha

    Topics Kähler–Ricci solitons, Einstein metrics, Kähler geometry

  4. Concavity of Perelman's -functional over the space of Kähler potentials

    Nefton Pali

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

  5. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci solitons, Einstein metrics, Kähler geometry

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

    Yasuhiro Nakagawa and Satoshi Nakamura

    Topics Kähler–Ricci solitons, Einstein metrics, Kähler geometry

  7. Kähler-Ricci solitons on horospherical variety

    François Delgove

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

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

    Guoqiang Wu and Shijin Zhang

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons

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) 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.