AI 中文总结
本文针对七、八维Hamilton常微分方程构造横截不变捏紧锥，修复Chen构造在n=7、8的失效步骤，结合相关结果扩展正各向同性曲率流形及Ricci流古老解的分类定理。
AI 中文摘要
我们针对n=7、8维的Hamilton常微分方程dR/dt = Q(R)构造了一族横截不变的捏紧锥，从而扩展了Brendle针对n≥12、Chen针对9≤n≤11维建立的捏紧估计。Chen的构造中有两个步骤在n=8时会真正失效：第一个是第二个锥族的估计，通过保留在维度通用论证中被舍弃的耦合关系得以修复；第二个是两个锥族的粘合，Chen的估计在n=8时存在确定幅度的失效，我们用新论证将粘合简化为弱PIC曲率算子锥上的单个线性不等式，并通过框架不等式的显式有限组合证明该不等式。在n=7时，另有两个步骤失效：第一个锥族背后的端点约化，以及我们粘合凭证背后的平均族部分。在重新平衡两个锥族的参数后，我们用直接多项式正性凭证取代前者，用基于第一Bianchi恒等式的不等式取代后者。所得捏紧估计结合Brendle的维度通用论证，在七维和八维给出不含非平凡不可压缩(n-1)维空间形式的正各向同性曲率紧流形的拓扑分类，扩展了Brendle针对n≥12的定理；结合Cho-Li与Brendle-Naff的曲率改进和分类结果，给出具有一致PIC的Ricci流非紧κ-非塌缩古老解的分类，扩展了Cho-Li针对n=4或n≥12的定理。
英文摘要
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 $n\geq 12$.
CommentsRevised version, 24 pages. Corrects Lemma 2.2, the entry property in Proposition 3.7, and several local formulas. The two cone families remain invariant in dimensions 7 and 8. The unproved 7-dimensional gluing step is removed; the gluing, full pinching, and classification results are now restricted to dimension 8. References have been updated
