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