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In this paper, we will study the asymptotic geometry of 4-dimensional steadygradient Ricci solitons under the condition that they dimension reduce to$3$-manifolds. We will show that such 4-dimensional steady gradient Riccisolitons either dimension reduce to a spherical space form$\mathbb{S}^3/\Gamma$ or weakly dimension reduce to the $3$-dimensional Bryantsoliton. We also show that 4-dimensional steady gradient Ricci solitonsingularity models with nonnegative Ricci curvature outside a compact seteither are Ricci-flat ALE $4$-manifolds or dimension reduce to $3$-dimensionalmanifolds. As an application, we prove that any steady gradientK\"{a}hler-Ricci soliton singularity models on complex surfaces withnonnegative Ricci curvature outside a compact set must be hyperk\"{a}hler ALERicc-flat $4$-manifolds.

On Four-dimensional Steady gradient Ricci solitons that dimension reduce