Paper · Feb 2, 2019

Constant -scalar curvature Kähler metric – formulation and foundational results

Eiji Inoue

math.DGarXiv:1902.00664v4
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Abstract

We introduce mu-scalar curvature for a K"ahler metric with a moment map mu and start up a study on constant mu-scalar curvature K"ahler metric as a generalization of both cscK metric and K"ahler-Ricci soliton and as a continuity path to extremal metric. We study some fundamental constraints to the existence of constant mu-scalar curvature K"ahler metric by investigating a volume functional as a generalization of Tian-Zhu's work, which is closely related to Perelman's W-functional. A new K-energy is studied as an approach to the uniqueness problem of constant mu-scalar curvature and as a prelude to new K-stability concept.

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Submitted
Feb 2, 2019
Last revised
Jan 27, 2021 (v4)
Subjects
math.DG

Comments: 44 pages. v4: Examples are added in section 6. Minor changes. v3: Section 5.2 is added. Chen-Tian formula of -Mabuchi functional is corrected (Proposition 4.2). (This change does not affect other results in this article. )

Ricci solitonsKähler geometryScalar curvatureStability & uniquenessKähler–Einstein & K-stability

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 The moduli space of Fano manifolds with Kähler-Ricci solitons (Eiji Inoue, 2018)4Inoue 2018 Existence of Kahler-Ricci solitons on smoothable Q-Fano varities (Yan Li, 2019)3Li 2019 Equivariant calculus on μ-character and μK-stability of polarized schemes (Eiji Inoue, 2020)1Inoue 2020 Entropies in μ-framework of canonical metrics and K-stability, I – Archimedean aspect: Perelman's W-entropy and μ-cscK metrics (Eiji Inoue, 2021)2Inoue 2021 The Ricci iteration towards cscK metrics (Kewei Zhang, 2023)6Zhang 2023 Formal structure of scalar curvature in generalized Kähler geometry (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2024)8Apostolov 2024 Pseudo-Calabi Flow on Toric Variety (Junhao Tian and Kai Zheng, 2026)7Tian 2026 Calabi flow on polarized Kähler surfaces (Haozhao Li, Bing Wang and Kai Zheng, 2026)5Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Equivariant calculus on -character and K-stability of polarized schemes

    Eiji Inoue

    Topics Kähler–Einstein & K-stability, Kähler geometry, Stability & uniqueness · Same author Eiji Inoue · Shared terms “foundational”, “Tian Zhu”, “volume functional”

  2. Entropies in -framework of canonical metrics and K-stability, I – Archimedean aspect: Perelman's W-entropy and -cscK metrics

    Eiji Inoue

    Topics Kähler–Einstein & K-stability, Kähler geometry, Stability & uniqueness · Same author Eiji Inoue · Shared terms “extremal metric”, “Perelman”

  3. Existence of Kahler-Ricci solitons on smoothable Q-Fano varities

    Yan Li

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

  4. The moduli space of Fano manifolds with Kähler-Ricci solitons

    Eiji Inoue

    Topics Kähler–Einstein & K-stability, Kähler geometry, Ricci solitons · Same author Eiji Inoue · Shared terms “moment map”

  5. Calabi flow on polarized Kähler surfaces

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Kähler geometry · Shared terms “cscK metric”, “extremal metric”, “continuity”

  6. The Ricci iteration towards cscK metrics

    Kewei ZhangWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Kähler geometry

  7. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows

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

  8. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Einstein & K-stability, Scalar curvature, Kähler geometry · Shared terms “moment map”

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 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.