Paper · Jan 27, 2021

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

Eiji Inoue

math.DGarXiv:2101.11197v2
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Abstract

This is the first in a series of two papers studying mu-cscK metrics and muK-stability, from a new perspective evoked from observations in arXiv:2004.06393 and in this first article. The first paper is about a characterization of mu-cscK metrics in terms of Perelman's W-entropy . We regard Perelman's W-entropy as a functional on the tangent bundle of the space of K"ahler metrics in a given K"ahler class . The critical points of turn out to be -cscK metrics. When , the supremum along the fibres gives a smooth functional on , which we call mu-entropy. Then -cscK metrics are also characterized as critical points of this functional, similarly as extremal metric is characterized as the critical points of Calabi functional. We also prove the W-entropy is monotonic along geodesics, following Berman–Berndtsson's subharmonicity argument. Studying the limit of the W-entropy, we obtain a lower bound of the mu-entropy. This bound is not just analogous, but indeed related to Donaldson's lower bound on Calabi functional by the extremal limit .

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Submitted
Jan 27, 2021
Last revised
Feb 24, 2022 (v2)
Subjects
math.DG, math.AG, math.CV

Comments: v2. 47 pages, Expressions are revised, comments by Laszlo Lempert are added, minimax interpretation of the result is additionally observed, a trailer on the second article was removed. Comments are very welcome!

Perelman entropyKähler geometryStability & uniquenessKähler–Einstein & K-stability

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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 The moduli space of Fano manifolds with Kähler-Ricci solitons (Eiji Inoue, 2018)6Inoue 2018 Constant μ-scalar curvature Kähler metric – formulation and foundational results (Eiji Inoue, 2019)2Inoue 2019 Equivariant calculus on μ-character and μK-stability of polarized schemes (Eiji Inoue, 2020)1Inoue 2020 Generalized Kähler-Ricci flow on toric Fano varieties (Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy, 2021)8Apostolov 2021 Kähler-Ricci flow for deformed complex structures (Gang Tian, Liang Zhang and Xiaohua Zhu, 2021)7Tian 2021 Weighted extremal metrics on blowups (Michael Hallam, 2023)4Hallam 2023 On the Existence of Weighted-cscK Metrics (Jiyuan Han and Yaxiong Liu, 2024)3Han 2024 A Spinorial Perelman's Functional: Critical Points and Gradient Flow (Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong, 2026)5Chow 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 “cscK metrics”, “observation”

  2. Constant -scalar curvature Kähler metric – formulation and foundational results

    Eiji Inoue

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

  3. On the Existence of Weighted-cscK Metrics

    Jiyuan Han and Yaxiong Liu

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

  4. Weighted extremal metrics on blowups

    Michael Hallam

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

  5. A Spinorial Perelman's Functional: Critical Points and Gradient Flow

    Tsz-Kiu Aaron Chow and Frederick Tsz-Ho Fong

    Topics Perelman entropy, Stability & uniqueness · Shared terms “critical points”

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

    Eiji Inoue

    Topics Kähler–Einstein & K-stability, Kähler geometry · Same author Eiji Inoue · Shared terms “studying”

  7. Kähler-Ricci flow for deformed complex structures

    Gang Tian, Liang Zhang and Xiaohua Zhu

    Topics Perelman entropy, Kähler–Einstein & K-stability, Kähler geometry · Shared terms “Donaldson”

  8. Generalized Kähler-Ricci flow on toric Fano varieties

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Perelman entropy, Kähler–Einstein & K-stability, 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.

Perelman entropy Perelman found that Ricci flow is a gradient flow once a measure is fixed. His 𝓕-functional and 𝓦-entropy increase along the flow; a lower bound… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) 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) 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. 6A.12
    Entropy, Information and DiffusionBook 6A · The Heat Equation and Its Relatives

    Entropy along the heat flow, the log-Sobolev inequality and Li–Yau: the ancestors of Perelman’s entropy.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6B.16
    Eigenvalue Problems and Perelman’s λBook 6B · Linear PDE: Characteristics, Waves and Regularity

    Rayleigh quotients and min–max, Faber–Krahn and Cheeger’s inequality, ground states of Schrödinger operators, spectral clustering, and the first eigenvalue of −4Δ + R that begins Perelman’s first preprint.