Paper · Oct 4, 2018

Pluripotential kahler-ricci flows

Vincent Guedj, Hoang Chinh Lu, Ahmed Zeriahi

math.CVarXiv:1810.02121
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Abstract

We develop a parabolic pluripotential theory on compact Kähler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Ampère equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the Kähler-Ricci flow on varieties with log terminal singularities.

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Submitted
Oct 4, 2018
Subjects
math.CV, math.AP, math.DG
Journal
Geom. Topol. 24 (2020) 1225-1296
DOI
10.2140/gt.2020.24.1225
Ricci flowKähler–Ricci flowSingularitiesKähler geometryWeak & singular flowsAnalytic minimal model program

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 Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension (P. Eyssidieux, V. Guedj and A. Zeriahi, 2016)2Eyssidieux 2016 The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)1Guedj 2018 Stability of solutions to complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)3Guedj 2018 Pluripotential Monge-Ampère flows in Big Cohomology Classes (Quang-Tuan Dang, 2021)6Dang 2021 Pluripotential Chern-Ricci Flows (Quang-Tuan Dang, 2022)4Dang 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)8Dang 2023 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)7Cho 2024 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)5Deruelle 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

    Topics Analytic minimal model program, Kähler geometry, Singularities · Same authors Vincent Guedj, Hoang Chinh Lu, Ahmed Zeriahi · Shared terms “pluripotential”, “complex Monge”, “Monge Ampère”

  2. Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension

    P. Eyssidieux, V. Guedj and A. Zeriahi

    Topics Analytic minimal model program, Kähler–Ricci flow, Weak & singular flows · Same authors Vincent Guedj, Ahmed Zeriahi · Shared terms “parabolic complex”, “degenerate parabolic”, “complex Monge”

  3. Stability of solutions to complex Monge-Ampere flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

    Topics Kähler geometry · Same authors Vincent Guedj, Hoang Chinh Lu, Ahmed Zeriahi · Shared terms “complex Monge”, “Monge Ampère”, “parabolic complex”

  4. Pluripotential Chern-Ricci Flows

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry, Singularities · Shared terms “pluripotential”, “terminal singularities”, “parabolic complex”

  5. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al.

    Topics Kähler–Ricci flow, Weak & singular flows, Kähler geometry · Same authors Vincent Guedj, Ahmed Zeriahi · Shared terms “compact Kähler”

  6. Pluripotential Monge-Ampère flows in Big Cohomology Classes

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry · Shared terms “pluripotential”, “Monge Ampère”, “compact Kähler”

  7. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “complex Monge”, “Ampère equation”, “Monge Ampère”

  8. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry, Singularities · Shared terms “complex Monge”, “parabolic complex”, “Ampère equation”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.