Paper · Apr 19, 2026

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

Jintao Ye, Xiaohua Zhu

math.DGarXiv:2604.17367v2
PDF Abstract page

As on arXiv

Abstract

We establish a Petersen-Wei type relative volume comparison theorem for weighted Riemannian manifolds under both -bounds of the Bakry-Émery Ricci curvature and the gradient of potential function. As an application, we give a modified proof for a volume comparison and monotonicity of Kähler-Ricci flow established in a recent work of Tian-Zhang-Zhang-Zhu-Zhu with improved estimate for error term.

Comments: We removed the volume growth condition in the main theorem in the first version of paper

Ricci flowKähler–Ricci flowKähler 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 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 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)4Tian 2022 Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition (Ling Wu, 2022)5Wu 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)1Wang 2023 A version of Bakry-Émery Ricci flow on a finite graph (Bobo Hua, Yong Lin and Tao Wang, 2024)3Hua 2024 Laplace comparison on Kähler Ricci flow and convergence (Gang Tian, Qi S. Zhang, Zhenlei Zhang et al., 2025)2Tian 2025 Rigidity of weighted manifolds via classification results for semilinear equations (Giulio Ciraolo, Alberto Farina and Troy Petitt, 2025)6Ciraolo 2025 Volume comparison for 3-manifolds with 2-Ricci curvature lower bound and Ricci flow (Shaochuang Huang and Zhuo Peng, 2026)7Huang 2026 Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds (Liding Huang and Xiaohua Zhu, 2026)8Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Xiaohua Zhu

  2. Laplace comparison on Kähler Ricci flow and convergence

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

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “comparison”, “Tian”

  3. A version of Bakry-Émery Ricci flow on a finite graph

    Bobo Hua, Yong Lin and Tao Wang PDF

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”

  4. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Xiaohua Zhu

  5. Time analyticity for the heat equation under Bakry-Émery Ricci curvature condition

    Ling Wu PDF

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”, “Zhang”

  6. Rigidity of weighted manifolds via classification results for semilinear equations

    Giulio Ciraolo, Alberto Farina and Troy Petitt PDF

    Topics Limit spaces & RCD · Shared terms “Émery Ricci”, “Bakry Émery”, “weighted Riemannian”

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

    Shaochuang Huang and Zhuo Peng PDF

    Topics Limit spaces & RCD · Shared terms “volume comparison”

  8. Convergence of Generalized Kähler–Ricci Flow on Toric Fano Manifolds

    Liding Huang and Xiaohua Zhu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.