Paper · Jan 30, 2026

Prescribed -curvature flow on the four-dimensional unit ball

Pak Tung Ho, Cheikh Birahim Ndiaye, Liming Sun, Heming Wang

math.DGWider flowsarXiv:2601.22934
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Abstract

In this paper, we study the prescribed -curvature problem on the unit ball of via the -curvature flow approach. By combining Ache-Chang's inequality with the Morse-theoretic approach of Malchiodi-Struwe, we establish existence results under strong Morse-type inequalities at infinity. As a byproduct of our argument, we also prove the exponential convergence of the -curvature flow on , starting from a -flat and minimal metric conformal to the standard Euclidean metric, to an extremal metric of Ache-Chang's inequality whose explicit expression was derived by Ndiaye-Sun.

Other curvature flows4-manifoldsLong-time existence & convergenceKähler–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 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 Static solutions to symplectic curvature flow in dimension four (Gavin Ball, 2023)2Ball 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)4Li 2023 Convergence rate of the Q-curvature flow (Pak Tung Ho and Sanghoon Lee, 2024)3Ho 2024 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)7Yang 2025 Inverse curvature flows for capillary hypersurfaces in the unit ball (Shujing Pan and Bo Yang, 2025)5Pan 2025 An improvement of regularity result for pseudo Calabi flow (Jingrui Cheng and Junhao Tian, 2026)6Cheng 2026 Capillary quermassintegral inequalities in the unit ball (Shujing Pan and Julian Scheuer, 2026)8Pan 2026 On global Chern-Calabi flows on compact complex surfaces (Li Ma and Li Sheng, 2026)1Ma 2026
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  1. On global Chern-Calabi flows on compact complex surfaces

    Li Ma and Li ShengWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “extremal metric”

  2. Static solutions to symplectic curvature flow in dimension four

    Gavin BallWider flows PDF

    Topics Kähler–Einstein & K-stability, 4-manifolds, Other curvature flows

  3. Convergence rate of the -curvature flow

    Pak Tung Ho and Sanghoon LeeWider flows PDF

    Topics Long-time existence & convergence, Other curvature flows · Same author Pak Tung Ho

  4. Calabi flow with bounded scalar curvature

    Haozhao Li, Linwei Zhang and Kai ZhengWider flows PDF

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

  5. Inverse curvature flows for capillary hypersurfaces in the unit ball

    Shujing Pan and Bo YangWider flows PDF

    Topics Other curvature flows · Shared terms “unit ball”, “Euclidean”

  6. An improvement of regularity result for pseudo Calabi flow

    Jingrui Cheng and Junhao TianWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows

  7. Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns

    Xiaorui Yang and Hao YuWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows

  8. Capillary quermassintegral inequalities in the unit ball

    Shujing Pan and Julian ScheuerWider flows PDF

    Topics Other curvature flows · Shared terms “unit ball”, “Euclidean”

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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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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)