Paper · May 11, 2026

On an invariant curvature cone along 4-dimensional Ricci flow

Hongting Ding, Shaochuang Huang, Zhuo Peng

math.DGarXiv:2605.10837v2
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Abstract

In this paper, we study 4-dimensional complete noncompact manifolds (M,g) satisfying Rm(g) via Ricci flow. Under the additional assumption of maximal volume growth, we prove topological and geometric gap theorems. We also study 4-dimensional complete manifolds satisfying a lower bound with respect to and obtain regularity results for Gromov-Hausdorff limits of complete volume non-collapsed manifolds satisfying such curvature lower bounds.

Comments: Some errors and typos of previous version have been corrected. 44 pages, all comments are welcome

Ricci flow(Non)collapsing4-manifoldsLimit spaces & RCDBoundaries & non-compact

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

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    Topics Boundaries & non-compact · Same authors Shaochuang Huang, Zhuo Peng

  3. Pseudolocality and strong uniqueness for complete Ricci flows

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

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  5. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

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    Topics 4-manifolds, (Non)collapsing, Limit spaces & RCD · Shared terms “Hausdorff limit”, “dimensional Ricci”, “Gromov Hausdorff”

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    Topics 4-manifolds, (Non)collapsing, Limit spaces & RCD · Shared terms “Hausdorff limit”, “Gromov”

  7. Short time existence for harmonic map heat flow with time-dependent metrics

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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) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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. 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.