Paper · Apr 7, 2021

Generalized Kähler-Ricci flow on toric Fano varieties

Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy

math.DGarXiv:2104.03268v2
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Abstract

We study the generalized Kähler-Ricci flow with initial data of symplectic type, and show that this condition is preserved. In the case of a Fano background with toric symmetry, we establish global existence of the normalized flow. We derive an extension of Perelman's entropy functional to this setting, which yields convergence of nonsingular solutions at infinity. Furthermore, we derive an extension of Mabuchi's -energy to this setting, which yields weak convergence of the flow.

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Submitted
Apr 7, 2021
Last revised
Jan 6, 2022 (v2)
Subjects
math.DG, math.AP

Comments: Final version to appear Trans. AMS. Significantly condensed from V1

Ricci flowKähler–Ricci flowPerelman entropyKähler geometryLong-time existence & convergenceKähler–Einstein & K-stability

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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.

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  1. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al.

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “global existence”, “initial data”

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

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    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Kähler geometry · Shared terms “toric Fano”, “generalized Kähler”, “symplectic type”

  3. Toric geometry of generalized Kähler-Ricci solitons

    Vestislav Apostolov, Giuseppe Barbaro, Jeffrey Streets et al.

    Topics Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “symplectic type”

  4. Formal structure of scalar curvature in generalized Kähler geometry

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler–Einstein & K-stability, Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”, “symplectic type”, “Mabuchi”

  5. The nondegenerate generalized Kähler Calabi-Yau problem

    Vestislav Apostolov and Jeffrey Streets

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets · Shared terms “generalized Kähler”, “weak convergence”, “global existence”

  6. Classification of generalized Kähler-Ricci solitons on complex surfaces

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    Topics Kähler–Ricci flow, Kähler geometry · Same authors Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

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

    Vestislav Apostolov, Jeffrey Streets and Yury Ustinovskiy

    Topics Kähler geometry · Same authors Vestislav Apostolov, Jeffrey Streets, Yury Ustinovskiy · Shared terms “generalized Kähler”

  8. Generalized Kähler-Ricci flow and the classification of nondegenerate generalized Kähler surfaces

    Jeffrey Streets

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Jeffrey Streets · Shared terms “generalized Kähler”

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 flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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)

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.