Paper · Nov 27, 2023

The Ricci iteration towards cscK metrics

Kewei Zhang

math.DGWider flowsarXiv:2311.15524v2
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Abstract

Motivated by the problem of finding constant scalar curvature Kähler metrics, we investigate a Ricci iteration sequence of Rubinstein that discretizes the pseudo-Calabi flow. While the long time existence of the flow is still an open question, we show that the iteration sequence does exist for all steps, along which the K-energy decreases. We further show that the iteration sequence, modulo automorphisms, converges smoothly to a constant scalar curvature Kähler metric if there is one, thus confirming a conjecture of Rubinstein from 2007 and extending results of Darvas–Rubinstein to arbitrary Kähler classes.

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Submitted
Nov 27, 2023
Last revised
May 1, 2025 (v2)
Subjects
math.DG, math.AP, math.DS

Comments: final version, to appear in Adv. Math

Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–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 2021, plus the canon.

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  3. An improvement of regularity result for pseudo Calabi flow

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  5. Calabi flow with bounded scalar curvature (II)

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  6. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “Kähler classes”, “Calabi flow”

  7. Calabi flow on polarized Kähler surfaces

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · Shared terms “Calabi flow”, “constant scalar”

  8. Ricci iterations of well-behaved Kähler metrics

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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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.