Compactness theorems for 4-dimensional gradient Ricci solitons

We prove compactness theorems for noncompact 4-dimensional shrinking and steady gradient Ricci solitons, respectively, satisfying: (1) every bounded open subset can be embedded in a closed 4-manifold with vanishing second homology group, and (2) are strongly κ-noncollapsed on all scales with respect to a uniform κ. These solitons are of interest because they are the only ones that can arise as finite-time singularity models for a Ricci flow on a closed 4-manifold with vanishing second homology group.