Rotational symmetry of self-similar solutions to the Ricci flow

TL;DR

This paper proves that any three-dimensional non-flat, -noncollapsed steady gradient Ricci soliton must be isometric to the Bryant soliton, confirming a longstanding conjecture in Ricci flow theory.

Contribution

It establishes the uniqueness of the Bryant soliton among three-dimensional steady gradient Ricci solitons, solving a problem posed by Perelman.

Findings

Any non-flat, -noncollapsed steady gradient Ricci soliton in 3D is isometric to the Bryant soliton.

The result confirms the conjecture about the rotational symmetry of such solitons.

It completes the classification of three-dimensional steady gradient Ricci solitons.

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.