Paper · Oct 1, 2026
Ricci flow on manifolds with positive isotropic curvature in all dimensions
As on arXiv
Abstract
We extend the classification of manifolds with positive isotropic curvature to all dimensions . This result also completes the classification of uniformly positive isotropic curvature on complete non-compact -noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.
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Pinching cones for positive isotropic curvature in dimensions seven and eight
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 12B.4Compactness and Universal Estimates for κ-Solutions
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.
- 11A.7Maximum Principles under Ricci Flow
Scalar and tensor maximum principles, and preserved curvature conditions.
- 11A.11Hamilton’s 1982 Theorem
Positive Ricci curvature in dimension 3 flows to a round metric.