Paper · Apr 30, 2020

Estimates for a geometric flow for the Type IIB string

Teng Fei, Duong H. Phong, Sebastien Picard, Xiangwen Zhang

math.DGarXiv:2004.14529
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Abstract

It is shown that bounds of all orders of derivative would follow from uniform bounds for the metric and the torsion 1-form, for a flow in non-Kähler geometry which can be interpreted as either a flow for the Type IIB string or the Anomaly flow with source term and zero slope parameter. A key ingredient in the proof is a formulation of this flow unifying it with the Ricci flow, which was recently found.

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Submitted
Apr 30, 2020
Subjects
math.DG, math.AP, math.CV

Comments: 20 pages

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)5Fei 2018 New curvature flows in complex geometry (Duong H. Phong, Sebastien Picard and Xiangwen Zhang, 2018)2Phong 2018 Unification of the Kähler-Ricci and Anomaly flows (Teng Fei and Duong H. Phong, 2019)1Fei 2019 Convergence of Hermitian manifolds and the Type IIB flow (Nikita Klemyatin, 2021)7Klemyatin 2021 Symplectic geometric flows (Teng Fei and Duong H. Phong, 2021)3Fei 2021 Geometric flows from unified string theories (Duong H. Phong, 2023)4Phong 2023 The Strominger System and Flows by the Ricci Tensor (Sébastien Picard, 2024)8Picard 2024 Flows of conformally coclosed G₂-structures with dilaton (Spiro Karigiannis, Sébastien Picard and Caleb Suan, 2025)6Karigiannis 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Unification of the Kähler-Ricci and Anomaly flows

    Teng Fei and Duong H. Phong

    Topics Kähler geometry · Same authors Teng Fei, Duong H. Phong · Shared terms “Anomaly flow”, “slope parameter”, “formulation”

  2. New curvature flows in complex geometry

    Duong H. Phong, Sebastien Picard and Xiangwen Zhang

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

  3. Symplectic geometric flows

    Teng Fei and Duong H. Phong

    Topics Ricci flow · Same authors Teng Fei, Duong H. Phong · Shared terms “geometric flows”

  4. Geometric flows from unified string theories

    Duong H. Phong

    Topics Kähler geometry · Same author Duong H. Phong · Shared terms “geometric flows”

  5. Parabolic Dimensional Reductions of 11D Supergravity

    Teng Fei, Bin Guo and Duong H. Phong

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

  6. 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”, “geometric flows”

  7. Convergence of Hermitian manifolds and the Type IIB flow

    Nikita Klemyatin

    Topics Ricci flow · Shared terms “Type IIb”

  8. The Strominger System and Flows by the Ricci Tensor

    Sébastien Picard

    Topics Kähler geometry · Same author Sebastien Picard · Shared terms “Anomaly flow”, “geometric flows”

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)

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.