Paper · May 17, 2017

Convergence of the Kähler-Ricci iteration

Tamás Darvas, Yanir A. Rubinstein

math.DGarXiv:1705.06253
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As on arXiv

Abstract

The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly. This article confirms this conjecture. As a special case, this gives a new method of uniformization of the Riemann sphere.

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Submitted
May 17, 2017
Subjects
math.DG, math.AP, math.CV
Journal
Analysis & PDE 12 (2019) 721-735
DOI
10.2140/apde.2019.12.721
Ricci flowEinstein metricsSurfacesKähler 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 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 Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics (Tamás Darvas, 2016)5Darvas 2016 Ricci iteration on homogeneous spaces (Artem Pulemotov and Yanir A. Rubinstein, 2016)1Pulemotov 2016 The Kähler-Ricci flow on pseudoconvex domains (Huabin Ge, Aijin Lin and Liangming Shen, 2018)6Ge 2018 On the Ricci iteration for homogeneous metrics on spheres and projective spaces (Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al., 2018)2Buttsworth 2018 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)7Guo 2020 Ricci iterations of well-behaved Kähler metrics (Andrea Loi and Giovanni Placini, 2023)3Loi 2023 The Ricci iteration towards cscK metrics (Kewei Zhang, 2023)4Zhang 2023 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)8Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci iteration on homogeneous spaces

    Artem Pulemotov and Yanir A. Rubinstein

    Topics Einstein metrics, Kähler geometry · Same author Yanir A. Rubinstein · Shared terms “Ricci iteration”, “discrete analogue”

  2. On the Ricci iteration for homogeneous metrics on spheres and projective spaces

    Timothy Buttsworth, Artem Pulemotov, Yanir A. Rubinstein et al.

    Topics Ricci flow · Same author Yanir A. Rubinstein · Shared terms “Ricci iteration”

  3. Ricci iterations of well-behaved Kähler metrics

    Andrea Loi and Giovanni Placini

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

  4. The Ricci iteration towards cscK metrics

    Kewei ZhangWider flows

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

  5. Metric geometry of normal Kähler spaces, energy properness, and existence of canonical metrics

    Tamás Darvas

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Same author Tamás Darvas

  6. The Kähler-Ricci flow on pseudoconvex domains

    Huabin Ge, Aijin Lin and Liangming Shen

    Topics Surfaces, Kähler–Einstein & K-stability, Einstein metrics

  7. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  8. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran

    Topics Kähler–Einstein & K-stability, Einstein metrics, 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 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… 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)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.