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

$\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons