176 21. PERELMAN'S PSEUDOLOCALITY THEOREM
900 (t) is fiat.^12 However, 900 (t) being fiat contradicts (21.43), which says
that J Rm 900 J(x 00 , 0) = 1. Hence we have proved that Claim 3' is true under
the assumption of Case 1.
Now we consider
Case 2 (Collapse - the rescaled injectivity radii tend to zero). Suppose
that the sequence of points and times {(xi,fi)} given in (21.21) satisfies(21.55) (^1) 11:n. sup (Q-1/2. i mJgi(fi). (-Xi )) =.^0
i-too
In this case we use the dilation factor Qi for (21.34) defined so that the
rescaled solution (Mf,§i (t)) in (21.34a) satisfies
(21.56) inj_gi(o) (xi) = 1.
This implies^13
(21.57)
- 1 A
_lim Qi Qi = oo.
i-too
Using the above two facts along with
J Rmgi J(x, t):::; 4QiQi - A-l on B§i(o) (-Xi, Ai --1/2 Al/2) [ a -^1 A J
10 Qi Qi x -2Qi Qi,^0
from (21.26) and A = lOO~coi ---+ oo, by applying Hamilton's (local) com-
pactness theorem to the pointed sequence
we obtain a subsequence converging to a smooth pointed solution
(21.58) (§oo (t), (xoo, 0)) on M~ x (-oo, OJ.In particular, the metric § 00 (0) on M 00 is well defined. Moreover, this limit
solution is complete and satisfies
(21.59)and
(^12) Since 900 (t) is a shrinking gradient soliton with extinction time t = 0, we have
S?P [Rm9 00 (t) [ = ~ S?P [Rm9 00 (to) [,
Moo Moo
so that limt-;o-supM 00 [Rm9 00 (t) [ = oo if supM 00 [Rm9 00 (to) [ -1-0.
(^131) n d d ee , QA i = lilJiii(o) • • (-Xi )-2 , sot h at
Qi Qi --1 A = (-1;2. Qi 1IlJ9i(o) • (xi) - )-^2 -+ oo.