Paper · Sep 26, 2019

Compact manifolds of dimension with positive isotropic curvature

Hong Huang

math.DGarXiv:1909.12265v22
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Abstract

We prove the following result: Let be a compact manifold of dimension with positive isotropic curvature. Then is diffeomorphic to a spherical space form, or the total space of an orbifiber bundle over or with generic fiber diffeomorphic to such that the total space admits a metric with positive isotropic curvature, where is a finite subgroup of acting freely on , and is the one dimensional closed orbifold with two singular points both with local group and with a closed interval, or a connected sum of a finite number of such manifolds. This extends a recent work of Brendle, and implies a conjecture of Schoen and a conjecture of Gromov in dimensions . The proof uses Ricci flow with surgery on compact orbifolds with isolated singularities.

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Submitted
Sep 26, 2019
Last revised
Jul 14, 2025 (v22)
Subjects
math.DG

Comments: 62 pages, added more details

Ricci flowSingularitiesSurgerySphere theorems

Its neighbourhood

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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. Classification of compact manifolds with positive isotropic curvature

    Hong Huang

    Topics Surgery, Sphere theorems · Same author Hong Huang · Shared terms “positive isotropic”, “isotropic curvature”, “compact manifolds”

  2. Open manifolds with uniformly positive isotropic curvature

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    Topics Surgery, Sphere theorems, Singularities · Same author Hong Huang · Shared terms “positive isotropic”, “isotropic curvature”, “isolated singularities”

  3. Ricci flow with surgery on manifolds with positive isotropic curvature

    S. Brendle

    Topics Surgery, Sphere theorems, Singularities · Shared terms “positive isotropic”, “isotropic curvature”, “compact manifolds”

  4. Shrinking Ricci solitons with positive isotropic curvature

    Keaton Naff

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”

  5. Four-dimentional Gradient Shrinking Solitons with Positive Isotropic Curvature

    Xiaolong Li, Lei Ni and Kui Wang

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”

  6. Ricci flow on manifolds with positive isotropic curvature in all dimensions

    Simon Brendle and Raphael Tsiamis

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”

  7. On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions

    Zhengnan Chen

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “Let”

  8. Manifolds with 1/4-pinched Curvature are Space Forms

    S. Brendle and R. M. SchoenThe canon

    Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “space form”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) 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. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

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

  2. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.

  3. 12B.12
    Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery

    The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).