Rotational symmetry of self-similar solutions to the Ricci flow

Abstract

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and \kappa-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman's first paper.Comment: Final version, to appear in Invent. Mat

Rotational symmetry of Ricci solitons in higher dimensions

“…This is a sequel to our earlier paper [4], in which we proved a uniqueness theorem for the three-dimensional Bryant soliton. Recall that the Bryant soliton is the unique steady gradient Ricci soliton in dimension 3, which is rotationally symmetric (cf.…”

“…[6]). In [4], it was shown that the threedimensional Bryant soliton is unique in the class of κ-noncollapsed steady gradient Ricci solitons: Theorem 1.1 (S. Brendle [4]). Let (M, g) be a three-dimensional complete steady gradient Ricci soliton which is non-flat and κ-noncollapsed.…”

“…In order to prove Theorem 1.2, we will adapt the arguments in [4]. While many arguments in [4] directly generalize to higher dimensions, there are several crucial differences.…”

“…While many arguments in [4] directly generalize to higher dimensions, there are several crucial differences. In particular, the proof of the roundness estimate in Section 2 is very different than in the three-dimensional case.…”

Gluing Eguchi‐Hanson Metrics and a Question of Page

“…It is an interesting question to classify ancient solutions and Ricci solitons. We refer to [10], [11], [12], [14] for some recent progress in this direction.…”