O(2)-symmetry of 3D steady gradient Ricci solitons

Abstract: For any 3D steady gradient Ricci soliton with positive curvature, we prove that it must be isometric to the Bryant soliton if it is asymptotic to a ray. Otherwise, it is asymptotic to a sector and hence a flying wing. We show that all 3D flying wings are O(2)-symmetric. Therefore, all 3D steady gradient Ricci solitons are O(2)-symmetric.