Paper · Feb 26, 2018

Relative volume comparison of Ricci Flow and its applications

Gang Tian, Zhenlei Zhang

math.DGarXiv:1802.09506v2
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Abstract

In this paper, we derive a relative volume comparison estimate along Ricci flow and apply it to studying the Gromov-Hausdorff convergence of Kähler-Ricci flow on a minimal manifold. This new estimate generalizes Perelman's no local collapsing estimate and can be regarded as an analogue of the Bishop-Gromov volume comparison for Ricci flow.

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Submitted
Feb 26, 2018
Last revised
Apr 17, 2018 (v2)
Subjects
math.DG

Comments: 28 pages; minor change in the proof of Lemma 3.1

Ricci flowKähler–Ricci flow(Non)collapsingKähler geometryLimit spaces & RCD

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 Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow (Jian Song, Gang Tian and Zhenlei Zhang, 2019)1Song 2019 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)6Jian 2023 Laplace comparison on Kähler Ricci flow and convergence (Gang Tian, Qi S. Zhang, Zhenlei Zhang et al., 2025)2Tian 2025 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)3Ye 2026 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 2026 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)8Huang 2026 Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I (Tongxin Xu and Zhenlei Zhang, 2026)5Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow

    Jian Song, Gang Tian and Zhenlei Zhang

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Same authors Gang Tian, Zhenlei Zhang · Shared terms “Gromov Hausdorff”

  2. Laplace comparison on Kähler Ricci flow and convergence

    Gang Tian, Qi S. Zhang, Zhenlei Zhang et al.

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Gang Tian, Zhenlei Zhang · Shared terms “comparison”

  3. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “volume comparison”, “relative volume”

  4. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “Gromov”, “Hausdorff”

  5. Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

    Tongxin Xu and Zhenlei Zhang

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Zhenlei Zhang

  6. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Same author Gang Tian · Shared terms “Gromov Hausdorff”, “Perelman”

  7. Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

    Tongxin Xu and Zhenlei Zhang

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same author Zhenlei Zhang

  8. Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow

    Shaochuang Huang and Zhuo Peng

    Topics (Non)collapsing, Limit spaces & RCD · Shared terms “volume comparison”, “along Ricci”, “Gromov Hausdorff”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.