1547671870-The_Ricci_Flow__Chow

266 9. TYPE I SINGULARITIES

and

IR~l2 = ~ [ (>, μ)2 + (,\ v)2 + (μ _ v)2] :::; ~ (>-2 + μ2 + v2) :::; 4 ,\2.

So if Iv\ :::; \,\ /2, we have

5 2 2 5 4 5 I

1

2

1

°

1

2

P'?:_ 2(3-5),\ (,- v) '?:. 8(3- 5),\ '?:. 96(3-5) Rm Rm'

while if \v i > I>-\ /2, we get

5 1

p > ,\2 (μ - v)2 + 2 (3 - 5) ,\2 (,\ - v)2 + 4>-2 (,\ - μ)2

'?:_ 2 (35- 5),\2 [(μ-v)2 + (,-v)2 + (,-μ)2]

5 2 0 2

'?:. 2 ( 3 _ 5 ) \Rm\ \Rml ,

b ecause 4 1 > 2 ( 38 _ 8 ).^0

LEMMA 9.13. Assume that for every c > 0, there exist T E [O, T) and

5 > 0 such that for every x E M^3 and t E [T, T), either

c

(9.6) \Rm (x, t)\ < T _ t

or

(9.7) \Rm - R (B ® B)\ > 5 IRm\

for every unit 2-form e at (x, t). Then F __, 0 uniformly as t __, T.

PROOF. By Lemma 9.11, F satisfies the differential inequality

(9.8)

where

H =:= (T - t~~c: [(B1 + B2 - G1) \Rm\ \R~l^2 - G2].

(R+ p)

The proof is in three steps.

We first show that H :::; O; by the maximum principle, this will prove

that F is nonincreasing. Observe that for every E > 0 and p < oo, there

exists T E [O, T) such that

1 c 1

B1 =Gip< ---= -G1

• 3T- t 3
  for all t E [T, T). Now if condition (9.6) holds at (x, t) with c = 1/ (3C2),
  then