Paper · May 6, 2019

Unification of the Kähler-Ricci and Anomaly flows

Teng Fei, Duong H. Phong

math.DGarXiv:1905.02274v2
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Abstract

A new formulation of the Anomaly flow in the case of vanishing slope parameter is given, where the dependence on the global section of the canonical bundle appears only in the initial data. This allows a natural unification of the Anomaly flow with the Kähler-Ricci flow.

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Submitted
May 6, 2019
Last revised
May 14, 2019 (v2)
Subjects
math.DG, hep-th, math-ph, math.AP
Journal
Differential Geometry, Calabi-Yau Theory, and General Relativity, Surveys in Differential Geometry Vol. 18, 89-104, 2018
DOI
10.4310/SDG.2018.v23.n1.a3

Comments: 16 pages, comments welcome! Some minor typos corrected in v2

Ricci flowKähler–Ricci flowKähler geometry

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)7Fei 2018 New curvature flows in complex geometry (Duong H. Phong, Sebastien Picard and Xiangwen Zhang, 2018)4Phong 2018 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)5Guo 2020 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)8Fei 2021 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)6Guo 2022 Geometric flows from unified string theories (Duong H. Phong, 2023)3Phong 2023 Flows of conformally coclosed G₂-structures with dilaton (Spiro Karigiannis, Sébastien Picard and Caleb Suan, 2025)2Karigiannis 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 Teng Fei, Duong H. Phong · Shared terms “Anomaly flow”, “slope parameter”, “formulation”

  2. Flows of conformally coclosed -structures with dilaton

    Spiro Karigiannis, Sébastien Picard and Caleb Suan

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Anomaly flow”

  3. Geometric flows from unified string theories

    Duong H. Phong

    Topics Kähler–Ricci flow, Kähler geometry · Same author Duong H. Phong

  4. New curvature flows in complex geometry

    Duong H. Phong, Sebastien Picard and Xiangwen Zhang

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

  5. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Ricci flow, Kähler geometry · Same author Duong H. Phong

  6. Diameter estimates in Kähler geometry

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

    Topics Kähler–Ricci flow, Kähler geometry · Same author Duong H. Phong

  7. Parabolic Dimensional Reductions of 11D Supergravity

    Teng Fei, Bin Guo and Duong H. Phong

    Topics Ricci flow · Same authors Teng Fei, Duong H. Phong

  8. Symplectic geometric flows

    Teng Fei and Duong H. Phong

    Topics Ricci flow · Same authors Teng Fei, 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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 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.

  3. 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.