Paper · Jan 15, 2017

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

Tat Dat Tô

math.CVarXiv:1701.04023v3
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Abstract

We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the Chern-Ricci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted Chern-Ricci flow.

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Submitted
Jan 15, 2017
Last revised
Jul 31, 2017 (v3)
Subjects
math.CV, math.AP, math.DG
Journal
Mathematische Annalen, 372 (2018), Issue 1-2, pp 699-741
DOI
10.1007/s00208-017-1574-7

Comments: 32 pages, final version, to appear in Math. Ann

Ricci flowGeneralized & G₂ flowsKähler geometry

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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 Weak Solutions of the Chern-Ricci flow on compact complex surfaces (Xiaolan Nie, 2017)7Nie 2017 Stability of solutions to complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)3Guedj 2018 Hermitian manifolds with quasi-negative curvature (Man-Chun Lee, 2018)6Lee 2018 The Chern-Ricci flow (Valentino Tosatti and Ben Weinkove, 2021)5Tosatti 2021 Pluripotential Chern-Ricci Flows (Quang-Tuan Dang, 2022)4Dang 2022 Singularities of the Chern-Ricci flow (Quang-Tuan Dang, 2023)2Dang 2023 Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds (Quang-Tuan Dang, 2026)1Dang 2026 On the Calabi estimate of geometric flows of Hermitian metrics (Marco Gallo and Luigi Vezzoni, 2026)8Gallo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

    Quang-Tuan Dang

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Hermitian manifolds”, “compact Hermitian”, “Ampère”

  2. Singularities of the Chern-Ricci flow

    Quang-Tuan Dang

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “complex Monge”, “Monge Ampère”

  3. Stability of solutions to complex Monge-Ampere flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

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

  4. Pluripotential Chern-Ricci Flows

    Quang-Tuan Dang

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Chern Ricci”, “Hermitian manifolds”, “complex Monge”

  5. The Chern-Ricci flow

    Valentino Tosatti and Ben Weinkove

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian”

  6. Hermitian manifolds with quasi-negative curvature

    Man-Chun Lee

    Topics Generalized & G₂ flows · Shared terms “Hermitian manifolds”, “Chern Ricci”, “compact Hermitian”

  7. Weak Solutions of the Chern-Ricci flow on compact complex surfaces

    Xiaolan Nie

    Topics Generalized & G₂ flows · Shared terms “Chern Ricci”, “Hermitian manifolds”, “compact Hermitian”

  8. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Hermitian”, “Chern Ricci”

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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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. 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.