Paper · May 30, 2019

Integral inequalities for holomorphic maps and applications

Yashan Zhang

math.DGarXiv:1905.13054v4
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Abstract

We derive some integral inequalities for holomorphic maps between complex manifolds. As applications, some rigidity and degeneracy theorems for holomorphic maps without assuming any pointwise curvature signs for both the domain and target manifolds are proved, in which key roles are played by total integration of the function of the first eigenvalue of second Ricci curvature and an almost nonpositivity notion for holomorphic sectional curvature introduced in our previous work. We also apply these integral inequalities to discuss the infinite-time singularity type of the Kähler-Ricci flow. The equality case is characterized for some special settings.

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Submitted
May 30, 2019
Last revised
Dec 4, 2020 (v4)
Subjects
math.DG

Comments: V4: Introduction section refined; accepted by Transactions of the American Mathematical Society

Ricci flowKähler–Ricci flowSingularitiesKähler geometryStability & uniquenessEigenvalues & spectrum

Its neighbourhood

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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 Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)5Nomura 2016 Infinite-time singularity type of the Kähler-Ricci flow (Yashan Zhang, 2017)1Zhang 2017 On a twisted conical Kähler-Ricci flow (Yashan Zhang, 2017)6Zhang 2017 Infinite-time singularity type of the Kähler-Ricci flow II (Yashan Zhang, 2018)2Zhang 2018 Local curvature estimates of long-time solutions to the Kähler-Ricci flow (Frederick Tsz-Ho Fong and Yashan Zhang, 2019)8Fong 2019 The spectral rigidity of Ricci soliton and Einstein-type manifolds (Ping Li, Xiaomei Sun and Anqiang Zhu, 2023)7Li 2023 Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities (Alexander Bednarek, 2024)4Bednarek 2024 Green's function estimates for compact Kähler manifolds and applications (Weiqi Zhang and Yashan Zhang, 2025)3Zhang 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinite-time singularity type of the Kähler-Ricci flow

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Yashan Zhang · Shared terms “singularity type”, “infinite time”, “time singularity”

  2. Infinite-time singularity type of the Kähler-Ricci flow II

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Yashan Zhang · Shared terms “singularity type”, “infinite time”, “time singularity”

  3. Green's function estimates for compact Kähler manifolds and applications

    Weiqi Zhang and Yashan Zhang

    Topics Eigenvalues & spectrum, Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang

  4. Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

    Alexander Bednarek

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “holomorphic maps”, “time singularity”

  5. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”

  6. On a twisted conical Kähler-Ricci flow

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “holomorphic”

  7. The spectral rigidity of Ricci soliton and Einstein-type manifolds

    Ping Li, Xiaomei Sun and Anqiang Zhu

    Topics Eigenvalues & spectrum, Kähler geometry, Stability & uniqueness · Shared terms “holomorphic sectional”, “sectional curvature”

  8. Local curvature estimates of long-time solutions to the Kähler-Ricci flow

    Frederick Tsz-Ho Fong and Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Yashan Zhang

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

In the guidebook

  1. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

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

  2. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.