Rotationally symmetric Ricci flow on ${\mathbb{R}}^{n+1}$

Abstract

We study the Ricci flow on ${\mathbb{R}}^{n+1}$, with $n\ge 2$, starting at some complete bounded curvature rotationally symmetric metric $g_0$. We first focus on the case where $({\mathbb{R}}^{n+1},g_0)$ 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, 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 $({\mathbb{R}}^{n+1},g_0)$ 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.

26 pages, final version. Accepted in Adv. Math