1547671870-The_Ricci_Flow__Chow

4. NECKLIKE POINTS IN FINITE-TIME SINGULARITIES 267

On the other hand, if condition (9.7) holds at (x, t), Lemma 9.12 implies
there is T/ = T/ ( o) such that
0
P 2: T/ 1 Rml^2 1Rml^2 ,

whence it follows that choosing E :::; ~ yields

B2 (1Rml IR~l^2 ) = C2s 1Rml^2 IR~l^2 :::; P = ~G2.

In either case, we have used at most 2/3 of G1 and at most 1 /2 of G2.
Therefore

(9.9) H < - (T- t)e: 3 [l ---IE Rml 1Rmlo^2 +P ] < 0.

• ( R + p) -e: 3 T - t -

We next improve estimate (9.8) Recall that either (9.6) or (9.7) holds at

(x, t). In the first case, the inequality R :::; Cs IRml implies there is T < T

such that (R(x, t) + p) (T - t) :::; 2cC3 fort E [T, T), whence we get

H:::; - (T - t)~R + p) rn IRml] :::; - 6c~3 IRml F

from (9.9). If (9.6) fails, then (9.7) holds and we have both conditions

IRm (x, t) I > c/ (T - t) and P 2: T/ IRml IRm^2 ° l^2. Thus by our choice of T , we

obtain

H:::; - (~T+-p;~~e: [TJ1Rml

2

IR~ 1

2

]

= - IRml2 F < - IRml c F < _ !] IRml F.

T/(R+p) - T/(R+p)(T- t) - 2Cs

Hence in either case we have

and thus

it F :::; b..F +

2

~;;) (\7 F, \7 R) - CI Rm I F.

Now we can show that F actually tends to zero uniformly. Since F

is bounded above, we may define Fmax (t) ~ sup Ms F (-, t). Recall that

0

in dimension 3, there is a constant A < oo such that 1Rml^2 :::; 1 Rml^2 <

A(R+p)^2. Let a> 0 be given, and consider any (x,t) with T:::; t < T.

Observe that (T - t) · (R + p) (x, t) :::; a only if

0

1 Rml^2

F ( x, t) = [ (T - t) ( R + p W 2 < Aae:.

(R+ p)

Thus if F (x, t) 2: Aae:, we must have

ca

IRml 2:c(R+p) > T - t