- EXISTENCE OF AN ASYMPTOTIC SHRINKER 113
Given an r > 0, if for any to E [r^2 , T) and any qo EM the solution g (t)
is not strongly A;-collapsed at (qo, to) at scaler, then we say that (M, g (t))
is weakly A;-noncollapsed at scale r (note the slight difference between
this and Definition 18.34).
Recall the weakened no local collapsing estimate and theorem (Theorem
8.24 and Theorem 8.26 in Part I).THEOREM 19.51 (Main estimate for weakened no local collapsing). Let(Mn, g (t)), t E [O, T), be a complete solution to the Ricci flow with T < oo
and suppose SUPMx[O,ti] IRml < oo for any ti < T. Then there exists c1
= c1 (n) E (0, !J depending only on n such that if for some A;l/n :S c 1 (n),the solution g (t) is strongly :A;-collapsed at (p, t) at scale r, where t* > ~
and r < ..;r;, then the reduced volume v of g ( 'T) ~ g ( t* - 'T) with basepoint
p* has the upper bound(19.59)where
E ~ A;l/nand¢ (2, n) ~ exp(~n(n-1)) ;^2 n-2 n-2 ( 1 )
(47rr^12 En + Wn_i(n - 2)^2 e--2-exp -2v2 ~.THEOREM 19.52 (Weakened no local collapsing). Let (Mn, g (t)), t E
[O, T), be a complete solution to the Ricci flow with T < oo. Suppose
(1) SUPMx[O,tiJ IRml < oo for any ti < T and(2) there exist rl > 0 and v1 > 0 such that Vol_q(o) B_g(o) (x, rl) :'.:'. v1 for
all x EM.
Then there exists A;> 0 depending only on rl, v1, n, T, and supMx[O,T/2]Rc g(t)
such that g (t) is weakly A;-noncollapsed at any point (p, t) E M x (T /2, T)
at any scale r < VTfi.
5.3. Proof of the existence of an asymptotic shrinker.
An important aspect of the geometry of A;-solutions is the existence of an
asymptotic gradient shrinking soliton, which is given by Perelman's Propo-
sition 11.2 in [152] (see also Theorem 8.32 in Part I).
THEOREM 19.53 (Existence of an asymptotic shrinking soliton for A;-solu-
tions). Let (Mn, g (t)), t E (-oo, OJ, be a A;-solution. Given a basepoint
(p,to), letr(t)~to-t andg(r)~g(to-r).
(1) For any sequence Ti ---+ oo, there exists qi EM such that
gg (q· i, r:) i < - C
for all i for some C < oo (we may take C = n/2).