The Rigidity of S ³ × R Under Ancient Ricci Flow

In this paper we generalize the neck-stability theorem of Kleiner–Lott (Acta Math 219: 65–134, 2014) to a special class of four-dimensional nonnegatively curved Type I κ-solutions, namely, those whose asymptotic shrinkers are the standard cylinder S ³ × R. We use this stability result to prove a rigidity theorem: if a four-dimensional Type I κ-solution with nonnegative curvature operator has the standard cylinder S ³ × R as its asymptotic shrinker, then it is exactly the cylinder with its standard shrinking metric.