Paper · Sep 7, 2016

On the extension and smoothing of the Calabi flow on complex tori

Hongnian Huang

math.DGWider flowsarXiv:1609.01834
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Abstract

In this paper, we continue to study the Calabi flow on complex tori. We develop a new method to obtain an explicit bound of the curvature of the Calabi flow. As an application, we show that when , the Calabi flow starting from a weak Kähler metric will become smooth immediately. It implies that in our settings, the weak minimizer of the Mabuchi energy is a smooth one.

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Submitted
Sep 7, 2016
Subjects
math.DG, math.AP
Other curvature flowsKähler geometryKähler–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 The Calabi flow with rough initial data (Weiyong He and Yu Zeng, 2017)4He 2017 Hermitian Calabi functional in complexified orbits (Jie He and Kai Zheng, 2023)8He 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)3Li 2023 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)1Li 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)2Cheng 2026 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)7Li 2026 Pseudo-Calabi Flow on Toric Variety (Junhao Tian and Kai Zheng, 2026)5Tian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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”

  2. 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”, “immediately”

  3. Calabi flow with bounded scalar curvature

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

  4. The Calabi flow with rough initial data

    Weiyong He and Yu ZengWider flows

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

  5. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows

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

  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”

  7. 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”

  8. Hermitian Calabi functional in complexified orbits

    Jie He and Kai ZhengWider flows

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

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.

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) Kähler–Einstein & K-stability A Fano manifold carries a Kähler–Einstein metric exactly when it is K-polystable, an algebraic condition (Chen–Donaldson–Sun, Tian). Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)