Remarks on noncompact steady gradient Ricci solitons

Assume M-n is a complete noncompact steady gradient Ricci soliton with positive Ricci curvature. First, by deriving a useful formula we characterize the condition of the scalar curvature and the potential function having a same level surface. Then, we assume the dimension n = 3 and characterize the rotational symmetry geometrically. Finally, for all dimensions n >= 3, we prove a dimension reduction result at spatial infinity under additional assumptions that M-n is a kappa-solution and the scalar curvature is O (1/r), where r is the distance function.