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

Abstract

By Perelman's $\mathcal L$-geodesic theory, we study the blow-down solutions on a noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M^n, g)$ $(n\ge 4)$ with nonnegative curvature operator and positive Ricci curvature away from a compact set of $M$. We prove that any compact split ancient solution of codimension one 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.