- THE COMPACTNESS OF /\:-SOLUTIONS 137
converges in the 000 Cheeger-Gromov topology to a limit solution(M~, 900 (t), X 00 ), t :S; 0.
If the limit (M~, 900 (t), x 00 ) is always contained in the collection, we say
that the collection is compact modulo scaling.
Note that
(20.22)
We may think of (xk, 0) as being the space-time origin, where the solution
9k (t) is normalized so that its scalar curvature at the origin is equal to
1. For this reason we call 9k (t) a curvature normalized sequence of
solutions.
REMARK 20.8. Note that an ancient solution with bounded curvature
either has positive scalar curvature or is Ricci fl.at. (See Lemma 2.18 in [45]
for example.)
Given K, > 0 and n E N - { 1}, let 9Jt~~n denote the collection of n-
dimensional K,-solutions with Harnack and l~t 9Jtn,/\: denote the collection of
n-dimensional K,-solutions. As we mentioned before,
on-;.J.J~n,K; c on-Harn ;.J.J~n,K; an d on-;.J.J~3,/\: = on-Harn ;.J.J~3,K; •
Perelman proved the following remarkable compactness result (see §11.7 of
[152] and §1.1 of [153]).
THEOREM 20.9 (Compactness of K,-solutions with Harnack). For any
K, > 0 and n :'.'.': 2, the collection 9Jt~~n is compact modulo scaling.
'
In dimension 3, by Proposition 19.44 we have the following.
COROLLARY 20.10 (Compactness of 3-dimensional K,-solutions). For any
fixed K, > 0 the collection of 3-dimensional K,-solutions is compact modulo
scaling.
By Theorem 19.56 and Corollary 20.10 we have the following. Fix K, > 0
and let 9Jt~c!t denote the collection of noncompact 3-dimensional K,-solutions
and let 9Jt~~!h denote the collection of nonspherical space form 3-dimensional
K,-solution~.
COROLLARY 20.11 (Compactness of 3-dimensional K,-solutions improved).(i) The collectionU
9Jtncpt
3,K; '
/\:>0
which is the collection of all 3-dimensional noncompact solutionswhich are K,-solutions for some K, > 0, is compact modulo scaling.
That is, for any sequence(M~,gk(t)) E LJ 9Jt~~!t
/\:>0