LAUSR

Abstract We study the Ricci flow on R n + 1 , with n ≥ 2 , starting at some complete bounded curvature rotationally symmetric metric g 0 . We first focus on the case where ( R n + 1 , g 0 ) does not contain minimal hyperspheres; we prove that if g 0 is asymptotic to a cylinder, then the solution develops a Type-II singularity and converges to the Bryant soliton after scaling, while if the curvature of g 0 decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotically flat initial data ( R n + 1 , g 0 ) containing a neck and we prove that if the neck is sufficiently pinched, in a precise way, the Ricci flow encounters a Type-I singularity.