1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

132 20. COMPACTNESS OF THE SPACE OF /\;-SOLUTIONS

for all (x,t) EM x (to,O] satisfying dg(tj(x,xo)::; iro.

(b) (N oncollapsed at one time) If we only assume

(20.12) Vol 9 (o) Bg(o)(xo, ro) 2:: wr 0 ,

then when -to 2:: Tor5, we have

(20.13) R(x, t) ::; Cr 02 + B(t + Tor5)-^1

for all (x,t) EM x (-Tor5,0J satisfying d 9 (t)(x,xo)::; iro.

REMARK 20.7. Note that statement (a) in the above proposition is in-

dependent of To.

We devote the rest of this section to proving this proposition. Since we

can rescale the solution g (t) to r 02 g (r5t), it suffices to prove the proposition

with ro = 1.

2.2.2. Proof of Proposition 20.6(a) with ro = 1.

Suppose that the proposition is false. Then there exist w 0 > 0, sequences

{Bk} -+ oo and {Ck} -+ oo, and a sequence of (not necessarily complete)

pointed solutions

{(Mk,gk(t),xok): t E [tok,O]},

where tok E (-oo, 0), to the Ricci flow with nonnegative curvature operator

such that

(1) the metric ball B 9 k(oj(Xok, 1) is compactly contained in Mk,

(2) at each time t E [tok, O]

(20.14) Vol 9 k(t)Bgk(tj(Xok, 1) 2:: wo for all k,

(3) there exist points (xk, tlk) satisfying d 9 k(tlk)(xk, Xok) < i, tlk E

( tok, 0], and (large curvature points)

(20.15) R 9 k (xk, tlk) > Ck+ Bk(t1k - tok)-^1.

With Proposition 18.25 regarding curvature control in a parabolic cylin-

der in mind, we take O" = ~' (Nn, h (t)) =(Mk, gk(t)), (to, ti] = (tok, tlk],

and p = xok and we define

(i)

Ak ~ Ak (~) ~ (Ck+ Bk(t1k - tok)-^1 ) ·min { ~ (t1k - tok), 23

1

04

}

and

( 11 "") D k ::::;=. mm. { lOO(n-l), A~/ 2. ear y k > 1or arge

2

i(tik-tok)Ck+Bk} Cl l A l .c k 1

and Ak-+ oo and Dk-+ oo.
By Lemma 18.24 and Proposition 18.25, there exist tk E (tok, tlk], with
tk - tok 2:: i (t1k - tok), and Yk E B 9 k(tk) (xok, !) such that
(I)

and