Paper · Sep 12, 2026

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

Shaochuang Huang, Zhuo Peng

math.DGarXiv:2609.13938
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Abstract

In this note, we study volume comparison for 3-manifolds with 2-Ricci curvature lower bound. We establish monotonicity and rigidity for geodesic sphere area and ball volume, and derive the pointed Gromov-Hausdorff precompactness under a uniform lower bound on the conjugate radius for 3-manifolds with 2-Ricci curvature lower bound. We also discuss the almost preservation of 2-Ricci curvature lower bound and the local non-collapsing propagation along Ricci flow. Finally, we point out a topological rigidity result on nonnegative scalar curvature.

Comments: 19 pages. All comments are welcome

Ricci flow(Non)collapsing3-manifoldsScalar curvatureLimit spaces & RCDStability & uniqueness

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On an invariant curvature cone along 4-dimensional Ricci flow

    Hongting Ding, Shaochuang Huang and Zhuo Peng PDF

    Topics (Non)collapsing, Limit spaces & RCD · Same authors Shaochuang Huang, Zhuo Peng · Shared terms “curvature lower”, “lower bound”, “Gromov Hausdorff”

  2. Ricci Flow under Kato-type curvature lower bound

    Man-Chun Lee PDF

    Topics (Non)collapsing, 3-manifolds, Scalar curvature · Shared terms “curvature lower”, “lower bound”

  3. Notes on scalar curvature lower bounds of steady gradient Ricci solitons

    Shota Hamanaka PDF

    Topics Limit spaces & RCD, Scalar curvature · Shared terms “curvature lower”, “lower bound”, “Gromov”

  4. A note on a diffeomorphism criterion via long-time Ricci flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics Scalar curvature · Same authors Shaochuang Huang, Zhuo Peng

  5. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng PDF

    Topics Stability & uniqueness · Same authors Shaochuang Huang, Zhuo Peng

  6. Strong Uniqueness for Instantaneously Complete Chern-Ricci Flow

    Shaochuang Huang and Zhuo Peng PDF

    Topics Stability & uniqueness · Same authors Shaochuang Huang, Zhuo Peng · Shared terms “comparison”

  7. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang PDF

    Topics Scalar curvature · Shared terms “curvature lower”, “lower bound”, “along Ricci”

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

    Jintao Ye and Xiaohua Zhu PDF

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

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) (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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

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

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.