2020/09/24 by Bennett Chow, Chow, Bennett, Yuxing Deng +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2009.11456
openalex publication_date 2020/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbS3/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 an 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 Ricc-flat 4-manifolds.