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 the condition that they dimension reduce to $3$-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either dimension reduce to a spherical space form $\mathbb{S}^3/\Gamma$ or weakly dimension reduce to the $3$-dimensional Bryant soliton. We also show that 4-dimensional steady gradient Ricci soliton singularity models with nonnegative Ricci curvature outside a compact set either are Ricci-flat ALE $4$-manifolds or dimension reduce to $3$-dimensional manifolds. As an application, we prove that any steady gradient K\"{a}hler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperk\"{a}hler ALE Ricc-flat $4$-manifolds.