44 28. SPECIAL ANCIENT SOLUTIONS
B 9 , 1 (o) (xi;, j^112 ) x [-j, O] is }-close to the corresponding subset of some ,,;-solution
(N},hj (t)) centered at some Yj E Nj.
Now, from Perelman 's compactness theorem for ,,;-solutions in dimension 3
(see Theorem 11.7 in Perelman [312] or the expository Corollary 20.10 in Part III)and since limj~oo R1i 1 (yj,0) = limj~oo R9, ] (xi;, 0) = 1, we have that (Nj, hj (t),
(yj, 0)) subconverges in the C^00 pointed Cheeger- Gromov sense to some ,,;-solution
(N!,h 00 (t), (y 00 ,0)), t E (-oo,O]. Since j --7 oo, we conclude that (M,gi 1 (t),(xi 1 , 0)) subconverges to (N 00 , h= (t), (y 00 , 0)), t E (-oo, O]. D
With this result, we may give thePROOF OF THEOREM 28 .20. Let (M~, 9 = (t)), t E (-oo,O], be a singularitymodel of a finite-time singular solution (M^3 , 9 (t)), t E [O, T), on a closed manifold.
Then, by definition there exists (x i , ti ) and K i --7 oo such that (M, 9i (t), (x i , 0) ) ,
where 9i (t) ~ Ki9 (t i + Ki-^1 t), converges in the C^00 pointed Cheeger- Gromov
sense to (M~, 900 (t), (x 00 , 0)), where x 00 E M 00 and 900 (t) is nonfl.at. Let R i ~
R 9 (Xi , ti). Since the limit
Hm Ki-l Ri = R 900 (xoo, 0) E (0, oo)
i~ooexists (here R 900 (x 00 , 0) > 0 follows from the strong maximum principle since 900 (t)
is nonfl.at), we a lso have that
converges in the C^00 pointed Cheeger- Gromov sense to (M 00 , g=(t),(x 00 , 0)), t E
(-oo,O], where § 00 (t) = c9 00 (c-^1 t) and c ~ R 900 (x 00 ,0). On the other hand,by Theorem 28. 22 , there exists a subsequence {ij} such that (M, gi 1 (t), (x i 1 ,0))
converges to a 11;-solution (N!,h 00 (t), (y 00 ,0)), t E (-oo,O]. It then follows
from the definition of Cheeger- Gromov convergence that (M=,§ 00 (t)) is isomet-
ric to (N 00 , h= ( t)) as solutions on the time interva l ( -oo, O]. Therefore 900 ( t) =
c^1 g= (ct), t E (-oo, OJ, is a ,,;-solution. D
2.3. Relation between singularity models and gradient solitons.
As evidenced below, the GRS are typical singularity models. However, although
there is a rich theory of singularity analysis in Ricci fl.ow by Hamilton and Perelman,
there are not many isometry classes of noncompact singularity models which are
known to exist. Besides ,,;-solutions , an important class of potential singularity
models consists of the GRS.
First we state the following important classification result for nonfl.at 3-dimen-
sional steadies due to Brendle.
THEOREM 28.23. Any K-noncollapsed nonfiat complete noncompact 3-dimen-
sional steady GRS must be the Bryant so liton.
Recall from Theorem 28.20 that any 3-dimensional singularity model must be a
,,;-solution. The following result concretely relates singularity models and gradient
solitons. It says that backward blow-down limits of ,,;-solutions, based at points
where the reduced distances are bounded , are nonfl.at shrinkers (see Chapter 7
of Part I for the definition of the reduced distance£).