1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. BACKWARD LIMIT OF ANCIENT A;-SOLUTION IS A SHRINKER 409

PROOF. First we shall prove that there exists o > 0 such that for any

T > 0,

(8.44)

for all (q, f) E B 9 ( 7 ) ( qn ~) x [A-^1 T, AT]. From (8.43), since our an-

cient solution has Rm~ 0 (in particular, R ~ 0)

1

\1 ve (q,f) I - ::; {343_.

g(T) v 4f

For any q E B g( 7 ) ( qn J E-^1 T) , let / ( s) be a minimal normal geodesic from

q 7 to q with respect to the metric g ( T). Since £ is locally Lipschitz, we have

I I

rdg(-r)(q,q(T)) I I

yf£(q,T)-yf£(qnT) ::; Jo \lyf£(t(s) ,T) g(T) ds

< f3.~= !3.

-V4r V4c

Since £(qn T) ::; ~'by the above estimate, we have

(8.45) R(q,T) <; (!%+ff)' for q E Bg(r) (qr,~).

From Lemma 7.65, we have for q EM

A-2 < _£(q,f) £ (q, T) 1 "f -TE [A-1 T,T, ]

£(q,f) £ (q, T) ::; A^2 1 "f -T E [ T, A T ].

(Note that the inequalities in the above two lines have the same form.)

Hence for q E B 9 ( 7 ) ( q 7 , J c^1 T) and f E [ A-^1 T, AT J ,

R(q,T) <; A

2

(!%+ff)

2

Now (8.44) is proved by choosing

o-

1

~ 3A

2

(!%+ff)

2

Now f R (q,f) ::; 5-^1 follows directly from (8.43) and (8.44); the lemma

is proved. D

For any sequence Ti --+ oo, consider the sequence of solutions

(Mn,gTi(B),qTi), (} E [A-^1 ,A].