Pinching cones for positive isotropic curvature in dimensions seven and eight

@misc{cho2026pinching,
  title = {{Pinching cones for positive isotropic curvature in dimensions seven and eight}},
  author = {Jae Ho Cho},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.26598},
  eprint = {2608.26598},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26598},
}

TY  - UNPB
TI  - Pinching cones for positive isotropic curvature in dimensions seven and eight
AU  - Jae Ho Cho
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.26598
UR  - https://arxiv.org/abs/2608.26598
AB  - We construct and prove the transversal invariance of two families of pinching cones for the Hamilton ODE d/dtR=Q(R) in dimensions n=7,8, thereby extending the pinching estimate established by Brendle for n≥ 12 and by Chen for 9≤ n≤ 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≥ 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≥ 12.
ER  -