On four-dimensional steady gradient Ricci solitons that dimension reduce

Abstract

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 solitons either strongly dimension reduce to a spherical space form S3/Γ 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 a further application, we prove that any steady gradient Kähler-Ricci soliton singularity models on complex surfaces with nonnegative Ricci curvature outside a compact set must be hyperkähler ALE 4-manifolds.