- NOTES AND COMMENTARY 155
§3. Originally, Theorem 20.9 was proved for noncompact ,.,;-solutions in
dimension 3 (see §11.7 of [152]). As was pointed out later by Perelman (see
§1.1 of [153]), one can enable the argument to include the compact case.
Finally we remark that, when considering any attempt to improve on
Theorem 20.9, the reader may wish to keep in mind the example of a nonflat
Ricci flat asymptotically locally Euclidean (ALE) manifold (Mn, g) with atleast quadratic curvature decay; in particular, there exists C < oo and
0 EM such that JRm 9 (O)[ > 0 and
[Rm (x)J d (x, 0)^2 ::::; C
for all x EM. Such a manifold is a static eternal solution to the Ricci flow
which is ,.,;-noncollapsed at all scales. (Nice examples are given by Kron-
heimer's (Ricci flat) hyper-Kahler ALE 4-manifolds (M^4 , g).) Consider any
sequence of points Xi -+ oo (with [Rm (xi)J =/::. 0) and consider the corre-
sponding sequence of pointed blown-down manifolds {(Mn, gi, xi)}, where
gi ~ JRm (xi) I g. Note that [Rm 9 i (xi) I = 1,
[Rm 9 i (O)[ = [Rm(xi)J-^1 [Rm 9 (O)[-+ oo,
and d 9 i (xi, 0) = [Rm (xi)J^112 d (xi, 0) ::::; VG. Hence such a sequence does
not have a subsequence which converges in the pointed Cheeger-Gromov
sense; that is, compactness fails.^24
·^24 We thank Bing Wang for suggesting these examples to us.