Abstract
We study the Ricci flow on Rn+1, with n≥2, starting at some complete bounded curvature rotationally symmetric metric g0. We first focus on the case where (Rn+1,g0) does not contain minimal hyperspheres; we prove that if g0 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 g0 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 (Rn+1,g0) 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.
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