Paper · Dec 3, 2021

Lower bound of modified -energy on a Fano manifold with degeneration for Kähler-Ricci solitons

Liang Zhang

math.DGarXiv:2112.02001
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Abstract

In this paper, we extend Tosatti's method to study the lower boundedness of modified -energy on a Fano manifold and apply this result to study the relative -stability of the deformation space of a Kähler Ricci soliton.

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Submitted
Dec 3, 2021
Subjects
math.DG
Ricci solitonsKähler–Ricci solitonsKähler geometryStability & uniqueness

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 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 Kähler-Ricci flow for deformed complex structures (Gang Tian, Liang Zhang and Xiaohua Zhu, 2021)1Tian 2021 Variational structure and uniqueness of generalized Kähler-Ricci solitons (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)6Apostolov 2021 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)4Tian 2022 A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons (Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang, 2022)8Cao 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)2Wang 2023 Generalized almost-Kähler-Ricci solitons (Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi, 2024)3Albanese 2024 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)5Nitta 2026 Uniqueness of the soliton vector field of a shrinking gradient Kähler-Ricci soliton (Ronan J. Conlon and Alix Deruelle, 2026)7Conlon 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness · Same author Liang Zhang · Shared terms “Fano manifold”, “modified”

  2. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci solitons, Kähler geometry, Ricci solitons · Shared terms “Fano manifold”, “degeneration”

  3. Generalized almost-Kähler-Ricci solitons

    Michael Albanese, Giuseppe Barbaro and Mehdi Lejmi

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

  4. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

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

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

    Yasufumi Nitta

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

  6. Variational structure and uniqueness of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Ricci solitons, Kähler geometry, Stability & uniqueness

  7. 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, Stability & uniqueness

  8. A Weil-Petersson Type Metric on the Space of Fano Kaehler-Ricci Solitons

    Huai-Dong Cao, Xiaofeng Sun and Yingying Zhang

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

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. 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.