1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. ALMOST ~-SOLUTIONS 131

In particular the dilated solutions gk(t) = Qk · 9k(Q"T;^1 t) satisfy

Vol fJk(o)BfJk(o) (xk, Ak)

(20.8) (Akt 2: Eo for all k.

Since Rm Bk(O) 2: 0, by the Bishop-Gromov volume comparison theorem we

have

Vol 9k(o)B9k(o) (xk, 1) 2: Eo

for all k such that Ak 2: 1.

· From the curvature bound JR9k (x, 0) J ::; 2 on BfJk(o) (xk, rkQk/^2 ) and

since rkQk^12 2: 1 for k large enough, we know by Proposition 19.3 (i.e., the

local injectivity radius estimate) that there exists 80 (c: 0 ) > 0 such that

injgk(o) (xk) 2: 80

for k large enough. We can apply Hamilton's local compactness theorem

(see Theorem 3.16 in Part I) to conclude that

(B9k(O) (xk, Qk^12 rk), gk(t), Xk), t E (tkQk, OJ,
converges to a solution (M~,g 00 (t),x 00 ), t E (-oo,O].
The limit is complete since Bgk(o)(xk, rk) is compactly contained in Mk
and Qk/^2 rk --+ oo. Clearly the limit is a nonflat solution of the Ricci flow and
has nonnegative bounded curvature operator (the boundedness of the cur-
vature follows from (20.5)). Now (20.8) and the Bishop volume comparison
theorem imply the asymptotic volume ratio has a positive lower bound:

(20.9) AVR (§ 00 (0)) 2: ~ > 0.

Wn

However this contradicts Corollary 20.2, which says that AVR (g 00 (0)) = O,

and the proposition is proved. D

2.2. Curvature bound under the noncollapsing assumption.
In this subsection we give a proof of Corollary 11.6 in Perelman [152]
regarding a curvature estimate.
2.2.1. Statement of the main result.

PROPOSITION 20.6 (Curvature estimate in noncollapsed balls). For ev-

ery w > 0 there exist B = B(w) < oo, C = C(w) < oo, and ro = ro(w) > 0

with the following properties: Let (Mn, g (t)), t E [to, O], where to E (-oo, 0),

be a (not necessarily complete) solution to the Ricci flow with nonnegative

curvature operator. Let ro > 0 be a constant, let xo E M, and suppose that

the metric ball Bg(o)(xo,ro) is compactly contained in M.

(a) (Noncollapsed on a time interval) If for each time t E [to, OJ,

(20.10) Vol 9 (t) Bg(t) (xo, ro) 2: wr 0 ,

then we have

(20.11) R(x, t) ::; Cr 02 + B(t - to)-^1