On four-dimensional steady gradient Ricci solitons that dimension reduce

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under condition that they dimension reduce to 3-manifolds. We show such either strongly a spherical space form S3/Γ or weakly 3-dimensional Bryant soliton. also soliton singularity models with nonnegative curvature outside compact set are Ricci-flat ALE 4-manifolds manifolds. As further application, prove any Kähler-Ricci on complex surfaces must be hyperkähler 4-manifolds.