1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. NO LOCAL·COLLAPSING VIA REDUCED VOLUME MONOTONICITY 405

PROOF OF PROPOSITION 8.28. Let c1 be chosen as in Lemma 8.29 and

let /'i,l/n :::; ci. If V E Bg,(p.,O) (o,.s-^1!^4 ) and 'f'Vi[o,cr2] is a minimal£-

geodesic, then by Lemma 8.29,

In the integral defining V1 (cr^2 ), it follows from (8.18) that we only need

to consider those vectors V E B 9 ,(p.,o) (o,.s-^1!^4 ) for which 'f'Vi[o,i;r2] is a

minimal £-geodesic. Hence

Finally we give an estimate for V2 ( cr^2 ) from above.

PROPOSITION 8.30. Under the assumptions of Theorem 8.24, we have

V2 (cr^2 ) :S:: Wn-1 (n - 2) n2

2

e_n2

2

exp {-

2

~}.

PROOF. By (8.23), we have

(47rcr2rn/2 e-£('Yv(cr2),cr2)£Jv (T):::; exp (-IVl~.(o)).