Paper · Jun 16, 2024

On the Existence of Weighted-cscK Metrics

Jiyuan Han, Yaxiong Liu

math.DGarXiv:2406.10939v3
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Abstract

In this paper, we prove that on a smooth Kähler manifold, the -coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in), e.g, cscK metrics, Kähler-Ricci solitons, -cscK metrics.

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Submitted
Jun 16, 2024
Last revised
Nov 17, 2025 (v3)
Subjects
math.DG, math.AP

Comments: This is the published version of the article

Ricci solitonsKähler–Ricci solitonsKähler geometryKähler–Einstein & K-stability

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 On canonical radial Kaehler metrics (Andrea Loi, Filippo Salis and Fabio Zuddas, 2021)3Loi 2021 Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type (Yasuhiro Nakagawa and Satoshi Nakamura, 2022)4Nakagawa 2022 Weighted K-stability for a class of non-compact toric fibrations (Charles Cifarelli, 2023)1Cifarelli 2023 Weighted extremal metrics on blowups (Michael Hallam, 2023)2Hallam 2023 Kähler-Ricci solitons on Fano threefolds with non-trivial moduli (Minghao Miao and Linsheng Wang, 2023)8Miao 2023 The Ricci iteration towards cscK metrics (Kewei Zhang, 2023)7Zhang 2023 Extremal Kähler–Ricci solitons on Fano manifolds are Kähler–Einstein (Yasufumi Nitta, 2026)5Nitta 2026 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)6Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Weighted K-stability for a class of non-compact toric fibrations

    Charles Cifarelli

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “weighted cscK”, “cscK metrics”, “weighted Mabuchi”

  2. Weighted extremal metrics on blowups

    Michael Hallam

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “cscK metrics”, “extremal”

  3. On canonical radial Kaehler metrics

    Andrea Loi, Filippo Salis and Fabio Zuddas

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “extremal”

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

    Yasuhiro Nakagawa and Satoshi Nakamura

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “Mabuchi”

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

    Yasufumi Nitta

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “extremal”

  6. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry · Shared terms “Mabuchi functional”

  7. The Ricci iteration towards cscK metrics

    Kewei ZhangWider flows

    Topics Kähler–Einstein & K-stability, Kähler geometry · Shared terms “cscK metrics”

  8. Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

    Minghao Miao and Linsheng Wang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci solitons, Kähler geometry

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) Kähler–Einstein & K-stability A Fano manifold carries a Kähler–Einstein metric exactly when it is K-polystable, an algebraic condition (Chen–Donaldson–Sun, Tian). 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.