Title: $ε$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons,
Abstract: 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 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 the
degeneration of the metrics on a family of complete non-compact four
dimensional 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.