  In [Cheeger-Tian 2005], Cheeger-Tian proved an $\epsilon$-regularity theorem
for $4$-dimensional Einstein manifolds without volume assumption. They
conjectured that similar results should hold for critical metrics with constant
scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional
manifolds and higher dimensional Einstein manifolds. In this paper we consider
all these problems. First, we construct counterexamples to the conjecture for
$4$-dimensional critical metrics and counterexamples to the conjecture for
higher dimensional Einstein manifolds. For $4$-dimensional shrinking Ricci
solitons, we prove an $\epsilon$-regularity theorem which confirms
Cheeger-Tian's conjecture with a universal constant $\epsilon$. For Ricci flow,
we reduce Cheeger-Tian's $\epsilon$-regularity conjecture to a backward
Pseudolocality estimate. By proving a global backward Pseudolocality theorem,
we can prove a global $\epsilon$-regularity theorem which partially confirms
Cheeger-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 finite
number of points.
