Paper · Nov 19, 2020

A Chern-Calabi flow on Hermitian manifolds

Xi Sisi Shen

math.DGWider flowsarXiv:2011.09683v4
View PDF View HTML arXiv page

As on arXiv

Abstract

We study an analogue of the Calabi flow in the non-Kähler setting for compact Hermitian manifolds with vanishing first Bott-Chern class. We prove a priori estimates for the evolving metric along the flow given a uniform bound on the Chern scalar curvature. If the Chern scalar curvature remains uniformly bounded for all time, we show that the flow converges smoothly to the unique Chern-Ricci-flat metric in the -class of the initial metric.

View PDF View HTML arXiv page
Submitted
Nov 19, 2020
Last revised
Feb 2, 2022 (v4)
Subjects
math.DG

Comments: 15 pages. Final version to appear in Journal of Geometric Analysis

Other curvature flowsRicci-flat & Calabi–YauKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability

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 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)4Li 2023 Hermitian metrics on complex non-Kähler manifolds (Daniele Angella, 2025)8Angella 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)3Li 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)6He 2025 Continuation and convergence for the higher dimensional conformal Chern–Calabi flow (Hongyu Chen, Li Ma and Li Sheng, 2026)1Chen 2026 On global Chern-Calabi flows on compact complex surfaces (Li Ma and Li Sheng, 2026)2Ma 2026 Pseudo-Calabi Flow on Toric Variety (Junhao Tian and Kai Zheng, 2026)7Tian 2026 Chern scalar curvature and finite-time singularities of the conformal Chern–Calabi flow (Hongyu Chen, Li Ma and Li Sheng, 2026)5Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Continuation and convergence for the higher dimensional conformal Chern–Calabi flow

    Hongyu Chen, Li Ma and Li ShengWider flows

    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Scalar curvature · Shared terms “Chern scalar”, “Hermitian manifolds”, “compact Hermitian”

  2. On global Chern-Calabi flows on compact complex surfaces

    Li Ma and Li ShengWider flows

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

  3. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows

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

  4. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Long-time existence & convergence · Shared terms “Calabi flow”, “uniformly bounded”

  5. Chern scalar curvature and finite-time singularities of the conformal Chern–Calabi flow

    Hongyu Chen, Li Ma and Li ShengWider flows

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

  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, Scalar curvature, Long-time existence & convergence · Shared terms “Calabi flow”, “converges smoothly”

  7. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Long-time existence & convergence · Shared terms “Calabi flow”, “flow converges”

  8. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella

    Topics Kähler–Einstein & K-stability, Scalar curvature, Kähler geometry · Shared terms “Chern”, “Hermitian manifolds”, “Kähler setting”

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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.