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

We analyse the blow-down solutions for n -dimensional ( n ≥ 4 ) noncompact κ -noncollapsed steady gradient Ricci solitons ( M , g ) with Rm ≥ 0 and Ric > 0 away from a compact set of M . As one of main results, we prove that the ( n - 1 ) -dimensional compact split limit ancient Ricci flows of type I and type II cannot occur simultaneously from the blow-down of ( M , g ) . Consequently, we prove that ( M , g ) with Rm ≥ 0 must be isometric to the Bryant Ricci soliton up to scaling, if there exists a sequence of normally rescaled Ricci flows of ( M , g ) , which converges subsequently to a family of shrinking quotient cylinders. The latter improves a previous result of Brendle.