Paper · Jun 29, 2018

New curvature flows in complex geometry

Duong H. Phong, Sebastien Picard, Xiangwen Zhang

math.DGarXiv:1806.11235v2
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Abstract

This is a survey of some of the recent developments on the geometric and analytic aspects of the Anomaly flow. It is a flow of -forms on a -fold which was originally motivated by string theory and the need to preserve the conformally balanced property of a Hermitian metric in the absence of a -Lemma. It has revealed itself since to be a remarkable higher order extension of the Ricci flow. It has also led to several other curvature flows which may be interesting from the point of view of both non-Kähler geometry and the theory of non-linear partial differential equations.

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Submitted
Jun 29, 2018
Last revised
Jul 9, 2018 (v2)
Subjects
math.DG, math.AP, math.CV

Comments: References and a short section added

Ricci flowOther curvature flowsKähler geometryPhysics

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 Parabolic Dimensional Reductions of 11D Supergravity (Teng Fei, Bin Guo and Duong H. Phong, 2018)6Fei 2018 Unification of the Kähler-Ricci and Anomaly flows (Teng Fei and Duong H. Phong, 2019)5Fei 2019 Estimates for a geometric flow for the Type IIB string (Teng Fei, Duong H. Phong, Sebastien Picard et al., 2020)1Fei 2020 Symplectic geometric flows (Teng Fei and Duong H. Phong, 2021)7Fei 2021 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)8Guo 2022 Geometric flows from unified string theories (Duong H. Phong, 2023)2Phong 2023 The Strominger System and Flows by the Ricci Tensor (Sébastien Picard, 2024)3Picard 2024 Flows of conformally coclosed G₂-structures with dilaton (Spiro Karigiannis, Sébastien Picard and Caleb Suan, 2025)4Karigiannis 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Estimates for a geometric flow for the Type IIB string

    Teng Fei, Duong H. Phong, Sebastien Picard et al.

    Topics Kähler geometry · Same authors Duong H. Phong, Sebastien Picard, Xiangwen Zhang · Shared terms “Anomaly flow”

  2. Geometric flows from unified string theories

    Duong H. Phong

    Topics Kähler geometry · Same author Duong H. Phong · Shared terms “complex geometry”, “string theory”, “Motivated”

  3. The Strominger System and Flows by the Ricci Tensor

    Sébastien Picard

    Topics Physics, Kähler geometry · Same author Sebastien Picard · Shared terms “Anomaly flow”, “balanced”

  4. Flows of conformally coclosed -structures with dilaton

    Spiro Karigiannis, Sébastien Picard and Caleb Suan

    Topics Kähler geometry · Same author Sebastien Picard · Shared terms “Anomaly flow”, “complex geometry”, “conformally”

  5. Unification of the Kähler-Ricci and Anomaly flows

    Teng Fei and Duong H. Phong

    Topics Kähler geometry · Same author Duong H. Phong · Shared terms “Anomaly flow”

  6. Parabolic Dimensional Reductions of 11D Supergravity

    Teng Fei, Bin Guo and Duong H. Phong

    Topics Physics · Same author Duong H. Phong · Shared terms “partial differential”, “differential equation”

  7. Symplectic geometric flows

    Teng Fei and Duong H. Phong

    Topics Ricci flow · Same author Duong H. Phong · Shared terms “complex geometry”, “Motivated”

  8. Diameter estimates in Kähler geometry

    Bin Guo, Duong H. Phong, Jian Song et al.

    Topics Kähler geometry · Same author Duong H. Phong

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Physics Ricci flow first appeared in physics, as the leading-order renormalization-group flow of the string sigma model (Friedan, 1980). Start with: Nonlinear Models in 2+ε Dimensions (Friedan, 1980)

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.