1547671870-The_Ricci_Flow__Chow

268 9. TYPE I SINGULARITIES

and hence

8F < b..F+ 2(1- c:) (\!F \!R)- Cea F.

ot - R+ p ' T-t

So the maximum principle implies that whenever Fmax (t) 2 Ad' at some

t E [T, T), we have

:t [( T - t)-Cca Fmax (t)] :::; 0

in the sense of limsup of forward difference quotients. In particular, we get

~F < CcAal+c ~ (3

dt max - T - t T - t

if F max ( t) 2 Aac. Since F is nonincreasing, this implies that for all t E
[T, T), we have

F max ( t) :::; max { Aac, F max ( T) + (3 log ~ = ~ }.

It follows that Fmax (t) :::; Aac fort sufficiently close to T. D

To finish the proof of Theorem 9.9, we need to verify that the solution
satisfies an injectivity radius estimate on the scale of its maximum curvature.
This follows from Perelman's No Local Collapsing result (Theorem 8.17).
Armed with this fact, we now proceed to the final step of the proof.

PROOF OF THEOREM 9 .9. We are finally ready to take a limit of dila-
tions about the singularity at time T < oo, even though we do not know
whether the singularity is of Type I or Type Ila. In either case, choose a
sequence (xi, ti) as in Section 3.1 of Chapter 8, taking care that the curva-
ture at each (xi, ti) is uniformly comparable to its maximum at time k For
co E (0, 1), set

Mf^0 ~ {x E M^3 : IRm(x,ti)l 2 comaxM3 IRm(-,ti)I}.

Then by Lemma 8.19, there is c 1 > 0 such that for all x E Mf^0 ,

coc1

!Rm (x, ti)I 2 --.

T- ti

That is , every point in Mf^0 is Type I coc 1 -essential. For such points, we

have (T - ti) (R (x, ti)+ p) 2 c > 0 by Lemma 9.10.

By the compactness theorem stated in Section 3 of Chapter 7 and Corol-
lary 9.7, the pointed sequence (M^3 ,gi (t) ,xi) limits to a complete ancient
solution (M~, g 00 (t), x 00 ) of the Ricci flow having bounded nonnegative
sectional curvature. Since we assumed there are eventually no Type I c-
essential c5-necklike points, the limit actually has positive sectional curva-
ture. By construction of that limit, there is for every point y E M~ a