Paper · Oct 4, 2018

The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows

Vincent Guedj, Hoang Chinh Lu, Ahmed Zeriahi

math.DGarXiv:1810.02122
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As on arXiv

Abstract

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Ampère equations, modelled on the Kähler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.

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Submitted
Oct 4, 2018
Subjects
math.DG, math.AP, math.CV
Ricci flowSingularitiesKähler geometryAnalytic minimal model program

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 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)5Eyssidieux 2016 Pluripotential kahler-ricci 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)2Dang 2021 Pluripotential Chern-Ricci Flows (Quang-Tuan Dang, 2022)4Dang 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)6Dang 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)8Deruelle 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pluripotential kahler-ricci 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. Pluripotential Monge-Ampère flows in Big Cohomology Classes

    Quang-Tuan Dang

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

  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 “Ampere flow”, “complex Monge”, “Monge Ampère”

  4. Pluripotential Chern-Ricci Flows

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry, Singularities · Shared terms “pluripotential”, “complex Monge”, “semi concave”

  5. 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 geometry · Same authors Vincent Guedj, Ahmed Zeriahi · Shared terms “complex Monge”, “Monge Ampère”, “parabolic complex”

  6. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang

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

  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 geometry, Singularities · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  8. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al.

    Topics Kähler geometry · Same authors Vincent Guedj, Ahmed Zeriahi

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) 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) Analytic minimal model program On a projective variety the Kähler–Ricci flow runs until its Kähler class stops being Kähler, a time read off from cohomology. Start with: Existence and degeneration of Kähler-Einstein metrics on minimal algebraic varieties of general type (Tsuji, 1988)

In the guidebook

  1. 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.

  2. 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.

  3. 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.