Paper · Jan 24, 2017

The Calabi flow with rough initial data

Weiyong He, Yu Zeng

math.DGWider flowsarXiv:1701.06943v2
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Abstract

In this paper, we prove that there exists a dimensional constant such that given any background Kähler metric , the Calabi flow with initial data satisfying \beginequation* \partial \bar \partial u_0 \in L^\infty (M) and (1- δ)ω< ω_u_0 < (1+δ)ω, \endequation* admits a unique short time solution and it becomes smooth immediately, where . The existence time depends on initial data and the metric . As a corollary, we get that Calabi flow has short time existence for any initial data satisfying \beginequation* \partial \bar \partial u_0 \in C^0(M) and ω_u_0 > 0, \endequation* which should be interpreted as a "continuous Kähler metric". A main technical ingredient is Schauder-type estimates for biharmonic heat equation on Riemannian manifolds with time weighted Hölder norms.

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Submitted
Jan 24, 2017
Last revised
Feb 22, 2017 (v2)
Subjects
math.DG, math.AP

Comments: We improved our previous result to initial L^\infty Kähler metrics with small L^\infty oscillation. We also added the application, see Theorem 1.7

Other curvature flowsKähler geometryWeak & singular flowsHarmonic maps & DeTurck trickKähler–Einstein & K-stability

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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 On the extension and smoothing of the Calabi flow on complex tori (Hongnian Huang, 2016)1Huang 2016 Compactness of Kähler metrics with bounds on Ricci curvature and I functional (Xiuxiong Chen, Tamás Darvas and Weiyong He, 2017)2Chen 2017 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)7Li 2023 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)3Li 2025 Twisted Calabi functional and twisted Calabi flow (Jie He and Haozhao Li, 2025)5He 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)6He 2025 An improvement of regularity result for pseudo Calabi flow (Jingrui Cheng and Junhao Tian, 2026)4Cheng 2026 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the extension and smoothing of the Calabi flow on complex tori

    Hongnian HuangWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “smooth immediately”, “become smooth”

  2. Compactness of Kähler metrics with bounds on Ricci curvature and I functional

    Xiuxiong Chen, Tamás Darvas and Weiyong HeWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same author Weiyong He · Shared terms “Calabi flow”

  3. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”

  4. An improvement of regularity result for pseudo Calabi flow

    Jingrui Cheng and Junhao TianWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “initial data”, “immediately”

  5. Twisted Calabi functional and twisted Calabi flow

    Jie He and Haozhao LiWider flows

    Topics Harmonic maps & DeTurck trick, Kähler–Einstein & K-stability, Other curvature flows · Shared terms “Calabi flow”, “short time”, “time existence”

  6. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

    Jie He and Haozhao LiWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “initial data”, “time existence”

  7. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”

  8. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”, “time existence”

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) Harmonic maps & DeTurck trick Ricci flow is only weakly parabolic, because it is invariant under diffeomorphisms. The DeTurck trick adds a Lie-derivative term that makes it… Start with: Harmonic Mappings of Riemannian Manifolds (Eells et al., 1964)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.