Paper · Oct 4, 2018

Stability of solutions to complex Monge-Ampere flows

Vincent Guedj, Hoang Chinh Lu, Ahmed Zeriahi

math.CVarXiv:1810.02123
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Abstract

We establish a stability result for elliptic and parabolic complex Monge-Ampère equations on compact Kähler manifolds, which applies in particular to the Kähler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

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Submitted
Oct 4, 2018
Subjects
math.CV, math.AP, math.DG
Ricci flowKähler geometryStability & uniqueness

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)3Eyssidieux 2016 Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds (Tat Dat Tô, 2017)8Tô 2017 The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)1Guedj 2018 Pluripotential kahler-ricci flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)2Guedj 2018 Pluripotential Monge-Ampère flows in Big Cohomology Classes (Quang-Tuan Dang, 2021)7Dang 2021 The L^(∞) estimate for parabolic complex Monge-Ampère equations (Qizhi Zhao, 2023)4Zhao 2023 Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds (Mathew George, 2024)6George 2024 Complex Monge-Ampère equations on Hermitian Manifolds: From bounded to smooth solutions (Papa Badiane, Chinh H. Lu and Ahmed Zeriahi, 2026)5Badiane 2026
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 Kähler geometry · Same authors Vincent Guedj, Hoang Chinh Lu, Ahmed Zeriahi · Shared terms “Ampere flow”, “complex Monge”, “Monge Ampère”

  2. Pluripotential kahler-ricci 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”

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

    P. Eyssidieux, V. Guedj and A. Zeriahi

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

  4. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao

    Topics Kähler geometry · Shared terms “complex Monge”, “Monge Ampère”, “parabolic complex”

  5. Complex Monge-Ampère equations on Hermitian Manifolds: From bounded to smooth solutions

    Papa Badiane, Chinh H. Lu and Ahmed Zeriahi

    Topics Ricci flow · Same author Ahmed Zeriahi · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  6. Complex Monge-Ampère equation for positive -forms on compact Kähler manifolds

    Mathew George

    Topics Kähler geometry · Shared terms “complex Monge”, “Monge Ampère”, “Ampère equation”

  7. Pluripotential Monge-Ampère flows in Big Cohomology Classes

    Quang-Tuan Dang

    Topics Kähler geometry · Shared terms “Ampere flow”, “Monge Ampère”, “complex Monge”

  8. Regularizing properties of Complex Monge-Ampère flows II: Hermitian manifolds

    Tat Dat Tô

    Topics Kähler geometry · Shared terms “Ampere flow”, “complex Monge”, “Monge Ampère”

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 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.