Paper · Feb 9, 2021

Pluripotential Monge-Ampère flows in Big Cohomology Classes

Quang-Tuan Dang

math.DGarXiv:2102.05189v3
View PDF View HTML arXiv page

As on arXiv

Abstract

We study pluripotential complex Monge-Ampère flows in big cohomology classes on compact Kähler manifolds. We use the Perron method, considering pluripotential subsolutions to the Cauchy problem. We prove that, under natural assumptions on the data, the upper envelope of all subsolutions is continuous in space and semi-concave in time, and provides a unique pluripotential solution with such regularity. We apply this theory to study pluripotential Kähler-Ricci flows on compact Kähler manifolds of general type as well as on stable varieties.

View PDF View HTML arXiv page
Submitted
Feb 9, 2021
Last revised
Jan 3, 2022 (v3)
Subjects
math.DG, math.AP, math.CV
DOI
10.1016/j.jfa.2021.109373

Comments: 47 pages, to appear in J. Funct. Anal

Ricci flowKähler geometryAnalytic 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 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)3Guedj 2018 Stability of solutions to complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)5Guedj 2018 Convergence of the weak Kähler-Ricci Flow on manifolds of general type (Tat Dat Tô, 2019)6Tô 2019 Pluripotential Chern-Ricci Flows (Quang-Tuan Dang, 2022)2Dang 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)4Dang 2023 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)8Cho 2024 Hermitian null loci (Quang-Tuan Dang, 2024)7Dang 2024
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 · Shared terms “pluripotential”, “Ampere flow”, “Monge Ampère”

  2. Pluripotential Chern-Ricci Flows

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry · Same author Quang-Tuan Dang · Shared terms “pluripotential”, “semi concave”, “Monge Ampère”

  3. Pluripotential kahler-ricci flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

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

  4. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry · Same author Quang-Tuan Dang · Shared terms “Monge Ampère”, “complex Monge”

  5. Stability of solutions to complex Monge-Ampere flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

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

  6. Convergence of the weak Kähler-Ricci Flow on manifolds of general type

    Tat Dat Tô

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

  7. Hermitian null loci

    Quang-Tuan Dang

    Topics Analytic minimal model program · Same author Quang-Tuan Dang

  8. 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 · Shared terms “Monge Ampère”, “compact Kähler”, “complex Monge”

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) 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. 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.