Paper · Nov 3, 2025

Minimal Degrees, Volume Growth, and Curvature Decay on Complete Kähler Manifolds

Yuang Shi

math.DGarXiv:2511.01263v3
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Abstract

We consider noncompact complete Kähler manifolds with nonnegative bisectional curvature. Our main results are: 1. Precise relations among refined minimal degree of polynomial growth holomorphic functions and holomorphic volume forms, (asymptotic volume ratio) and (average of scalar curvature decay) are established. 2. The Lyapunov asymptotic behavior of the Kähler-Ricci flow can be described in terms of polynomial growth holomorphic functions. This provides a unifying perspective that bridges the two distinct proofs of Yau's uniformization conjecture by Liu and Chau-Lee-Tam. These resolve two conjectures made by Yang.

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Submitted
Nov 3, 2025
Last revised
Apr 25, 2026 (v3)
Subjects
math.DG
Journal
Int. Math. Res. Not. IMRN 2026, rnag080

Comments: 19 pages, version for publication, revised after referee report

Ricci flowKähler–Ricci flowSurfacesKähler geometryScalar curvatureBoundaries & non-compact

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 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 The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers (Fei He and Jianyu Ou, 2024)2He 2024 Dimension estimate and existence of holomorphic sections with polynomial growth on gradient Kähler Ricci shrinkers (Fei He and Jianyu Ou, 2024)3He 2024 Uniqueness of asymptotically conical Kähler-Ricci flow (Longteng Chen, 2025)5Chen 2025 Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker (Jiangtao Li, 2025)1Li 2025 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)6Di Nezza 2026 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)7Zhang 2026 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)4Di Nezza 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)8Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite generation of the ring of holomorphic functions with polynomial growth on the Kähler-Ricci shrinker

    Jiangtao Li

    Topics Scalar curvature, Boundaries & non-compact, Kähler geometry · Shared terms “holomorphic functions”, “polynomial growth”

  2. The dimension of polynomial growth holomorphic functions and forms on gradient Kähler Ricci shrinkers

    Fei He and Jianyu Ou

    Topics Kähler geometry · Shared terms “growth holomorphic”, “holomorphic functions”, “polynomial growth”

  3. Dimension estimate and existence of holomorphic sections with polynomial growth on gradient Kähler Ricci shrinkers

    Fei He and Jianyu Ou

    Topics Boundaries & non-compact, Kähler geometry · Shared terms “polynomial growth”, “holomorphic”

  4. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu

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

  5. Uniqueness of asymptotically conical Kähler-Ricci flow

    Longteng Chen

    Topics Kähler–Ricci flow, Scalar curvature, Boundaries & non-compact · Shared terms “curvature decay”

  6. Kähler-Ricci Flow: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu

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

  7. Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold

    Lei Zhang and Zhenlei Zhang

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry

  8. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song

    Topics Kähler–Ricci flow, Scalar curvature, 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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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.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.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.