#### Vol. 303, No. 1, 2019

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Compactness theorems for 4-dimensional gradient Ricci solitons

### Yongjia Zhang

Vol. 303 (2019), No. 1, 361–384
##### Abstract

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 $\kappa$-noncollapsed on all scales with respect to a uniform $\kappa$. 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.

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