Paper · Aug 18, 2026

Small Normal Curvature and Three-Manifold Topology

Tsz-Kiu Aaron Chow, Jingbo Wan

math.DGarXiv:2608.18002v2
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Abstract

For , we prove that every smooth immersion satisfies , with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Euclidean unit ball with is diffeomorphic to , , or . All three possibilities occur, while forces . These results answer a question of Petrunin and prove a conjecture of Chodosh–Li concerning the normal curvature of three-manifolds. The key intrinsic input is the strict scalar–systolic inequality for every spherical three-space form with . Its proof uses systolic monotonicity along Ricci flow with surgery. This strict inequality complements the sharp scalar–systolic inequality for of Bray–Brendle–Eichmair–Neves.

Comments: New results added: a classification of closed orientable 3-manifolds, a sharp sphere theorem, and a Ricci flow scalar–systolic gap

Ricci flowSurgery3-manifoldsSphere theorems

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 Ricci flow with surgery on three-manifolds (Grisha Perelman, 2003)6Perelman 2003 Notes on Perelman's papers (Bruce Kleiner and John Lott, 2006)3Kleiner 2006 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)5Kleiner 2014 Classification of compact manifolds with positive isotropic curvature (Hong Huang, 2023)2Huang 2023 Open manifolds with uniformly positive isotropic curvature (Hong Huang, 2023)4Huang 2023 On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces (Nurlan Abiev, 2023)8Abiev 2023 Rigidity of area non-increasing maps (Man-Chun Lee, Luen-Fai Tam and Jingbo Wan, 2023)1Lee 2023 Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery (John Hughes, 2025)7Hughes 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rigidity of area non-increasing maps

    Man-Chun Lee, Luen-Fai Tam and Jingbo Wan PDF

    Topics Sphere theorems · Same author Jingbo Wan · Shared terms “immersion”, “Brendle”

  2. Classification of compact manifolds with positive isotropic curvature

    Hong Huang PDF

    Topics Surgery, Sphere theorems · Shared terms “space forms”, “diffeomorphic”, “Brendle”

  3. Notes on Perelman's papers

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  4. Open manifolds with uniformly positive isotropic curvature

    Hong Huang PDF

    Topics Surgery, Sphere theorems · Shared terms “diffeomorphic”

  5. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon PDF

    Topics Surgery, 3-manifolds

  6. Ricci flow with surgery on three-manifolds

    Grisha PerelmanThe canon PDF

    Topics Surgery, 3-manifolds

  7. Towards a Taub-Bolt to Taub-NUT via Ricci flow with surgery

    John Hughes PDF

    Topics Surgery

  8. On the dynamics of a three-dimensional differential system related to the normalized Ricci flow on generalized Wallach spaces

    Nurlan Abiev PDF

    Topics Sphere theorems, 3-manifolds

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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 11B.7
    Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities

    What was proved, what was missing, and where Perelman came in.

  3. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.