Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set

Abstract

Let $(M^n,g)$ $(n\ge 4)$ be a complete noncompact $\kappa$-noncollapsed steady Ricci soliton with $\rm{Rm}\geq 0$ and $\rm{Ric}> 0$ away from a compact set $K$ of $M$. We prove that there is no any $(n-1)$-dimensional compact split limit Ricci flow of type I arising from the blow-down of $(M, g)$, if there is an $(n-1)$-dimensional noncompact split limit Ricci flow. Consequently, the compact split limit ancient flows of type I and type II cannot occur simultaneously from the blow-down. As an application, we prove that $(M^n,g)$ with $\rm{Rm}\geq 0$ must be isometric the Bryant Ricci soliton up to scaling, if there exists a sequence of rescaled Ricci flows $(M,g_{p_i}(t); p_i)$ of $(M,g)$ converges subsequently to a family of shrinking quotient cylinders.