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In [Cheeger-Tian 2005], Cheeger-Tian proved an $\epsilon$-regularity theoremfor $4$-dimensional Einstein manifolds without volume assumption. Theyconjectured that similar results should hold for critical metrics with constantscalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensionalmanifolds and higher dimensional Einstein manifolds. In this paper we considerall these problems. First, we construct counterexamples to the conjecture for$4$-dimensional critical metrics and counterexamples to the conjecture forhigher dimensional Einstein manifolds. For $4$-dimensional shrinking Riccisolitons, we prove an $\epsilon$-regularity theorem which confirmsCheeger-Tian's conjecture with a universal constant $\epsilon$. For Ricci flow,we reduce Cheeger-Tian's $\epsilon$-regularity conjecture to a backwardPseudolocality estimate. By proving a global backward Pseudolocality theorem,we can prove a global $\epsilon$-regularity theorem which partially confirmsCheeger-Tian's conjecture for Ricci flow. Furthermore, as a consequence of the$\epsilon$-regularity, we can show by using the structure theorem of Naber-Tian\cite{NaTi} that a collapsed limit of shrinking Ricci solitons with bounded$L^2$ curvature has a smooth Riemannian orbifold structure away from a finitenumber of points.

$\epsilon$-regularity for shrinking Ricci solitons and Ricci flows