{
  "abstract": "We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE $\\frac{\\mathrm{d}}{\\mathrm{d}t}R=Q(R)$ in dimensions $n=7,8$, thereby extending the pinching estimate established by Brendle for $n\\geq 12$ and by Chen for $9\\leq n\\leq 11$. In dimension $n=8$, two steps in Chen's construction genuinely fail. The first, an estimate for the second cone family, is repaired by retaining a coupling discarded in the dimension-general argument. The second is the gluing of the two cone families, for which Chen's estimate fails at $n=8$ by a definite margin. We replace it with a new argument that reduces the gluing to a single linear inequality on the cone of weakly PIC curvature operators, and prove this inequality by an explicit finite combination of frame inequalities. In dimension $n=7$, the two separate cone-invariance results are proved, but the gluing step is not established here. Accordingly, the resulting full pinching estimate and classification statements are restricted to dimension $n=8$. The $n=8$ pinching estimate, together with dimension-free arguments of Brendle, yields the topological classification in dimension eight of compact manifolds with positive isotropic curvature that contain no nontrivial incompressible $(n-1)$-dimensional space forms, extending a theorem of Brendle from $n\\geq 12$. Together with curvature-improvement and classification results of Cho--Li and Brendle--Naff, it also yields the classification of noncompact $κ$-noncollapsed ancient solutions to the Ricci flow with uniformly PIC, extending a theorem of Cho--Li from $n=4$ or $n\\geq 12$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.26598",
  "author_total": 1,
  "authors": [
    {
      "name": "Jae Ho Cho"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/28eb4148p9cry.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.26598",
  "fulltext_url": null,
  "id": "28eb4148p9cry",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.26598",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7204600036",
  "pdf_url": "https://arxiv.org/pdf/2608.26598",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/28eb4148p9cry.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.26598",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.26598"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Pinching cones for positive isotropic curvature in dimensions seven and eight",
  "title_text": "Pinching cones for positive isotropic curvature in dimensions seven and eight",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/28eb4148p9cry",
  "venue": "arXiv",
  "versions": []
}