Paper · Apr 5, 2023

Geometric flows from unified string theories

Duong H. Phong

math.DGarXiv:2304.02533
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Abstract

A survey of new geometric flows motivated by string theories is provided. Their settings can range from complex geometry to almost-complex geometry to symplectic geometry. From the PDE viewpoint, many of them can be viewed as intermediate flows between the Ricci flow and the Kähler-Ricci flow, albeit often coupled to flows of additional fields. In particular, a survey is given of joint works of the author with Tristan Collins, Teng Fei, Bin Guo, Sebastien Picard, and Xiangwen Zhang.

Comments: Contribution to Surveys in Differential Geometry, "Forty Years of Ricci flow", edited by H.D. Cao, R. Hamilton, and S.T. Yau

Ricci flowKähler–Ricci flowKähler geometry

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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 2021, 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 Symplectic geometric flows (Teng Fei and Duong H. Phong, 2021)1Fei 2021 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)2Guo 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)6Wu 2022 The L^(∞) estimate for parabolic complex Monge-Ampère equations (Qizhi Zhao, 2023)4Zhao 2023 Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold (Tomoyuki Hisamoto, 2024)7Hisamoto 2024 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)8Tosatti 2024 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)5Di Nezza 2026 Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation (Randall R. Van Why, 2026)3Van Why 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Symplectic geometric flows

    Teng Fei and Duong H. Phong PDF

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

  2. Diameter estimates in Kähler geometry

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

    Topics Kähler–Ricci flow, Kähler geometry · Same author Duong H. Phong · Shared terms “PDE”

  3. Kähler-Ricci Flow is Symplectic Deflation plus an Exact Symplectic Deformation

    Randall R. Van Why PDF

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

  4. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao PDF

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

  5. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu PDF

    Topics Kähler–Ricci flow, Kähler geometry

  6. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry

  7. Quantization of the Kähler-Ricci flow and optimal destabilizer for a Fano manifold

    Tomoyuki Hisamoto PDF

    Topics Kähler–Ricci flow, Kähler geometry

  8. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti PDF

    Topics Kähler–Ricci flow, Kähler geometry

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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… 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.