Inserted: 24 jul 2011
Last Updated: 24 jan 2014
Journal: Calc. Var. Partial Differential Equations
In this paper we prove that any $n$-dimensional ($n> 3$) complete Bach-flat gradient steady Ricci solitons with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton.