Uniqueness of standard solutions in the work of Perelman

The short time existence of Ricci flow on complete noncompact Riemannian manifolds with bounded curvature is proven by Shi [Sh1]. The uniqueness of such solutions is a difficult problem. Hsu studies this problem in dimension two [Hs], otherwise there is not any result about this problem. In the fundamental paper [Pe2] Perelman discussed a special family of solutions of Ricci flow on R, the so-called standard solutions, the solutions are used to construct the geometric-topological surgeries and their uniqueness are used to construct the longtime existence and to study the properties of the Ricci flow with surgery. A particular nice feature about these solutions is that at space infinity these solutions are asymptotic to round infinity cylinder. In [Pe2] §2 Perelman gives a proof of the uniqueness of the standard solutions. The idea is to reduce the Ricci flow equation to