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A closed four-dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small   $L^{2}$  -norm of the curvature. In this paper, we localize this fact in the case of gradient shrinking Ricci solitons by proving an   $\varepsilon $  -regularity theorem, thus confirming a conjecture of Cheeger–Tian [ 20 ]. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four-dimensional gradient shrinking Ricci solitons  without  a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons