No compact split limit Ricci flow of type II from the blow-down

By Perelman's L-geodesic theory, we study the blow-down solutions on a noncompact κ-noncollapsed steady gradient Ricci soliton (Mⁿ, g) (n≥ 4) with nonnegative curvature operator and positive Ricci curvature away from a compact set of M. We prove that any $(n-1)$-dimensional compact split ancient solution from the blow-down of $(M, g)$ is of type I. The result is a generalization of our previous work from $n=4$ to any dimension.