1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. BACKWARD LIMIT OF ANCIENT ~-SOLUTION IS A SHRINKER 413

4.4. The limit of the reduced volume. Note that the limit g 00 (e) is

defined for e E [A-1, A J. Instead of considering the reduced volume of this

limit, we define the function

which will play the role of the reduced volume; formally this is the reduced
volume using the limit function £ 00 • The fact that V 00 (e) is finite follows
from the following lemma. ·

LEMMA 8.38.

(i) We have

(ii) V 00 ( e) is a constant contained in ( 0, 1).

(iii) For any 'ljJ(e) which has compact support in (A-^1 ,A),

(8.50). L~1 JM= (41Te)-n/2 e-e=(q,e)'l/J'(e~dμg=(e)(q)de

= !A v~(e)1fJ'(e)de = o,

JA-l

where 'ljJ'(e) = d'ljJ/de.

PROOF. (i) Let Ti be a subsequence such that both g 7 i and fi(q, e) con-
verge. By (8.23), we have

( 41TTie)-~l2e-e('YV(rie),rie) £ Jv (Tie) :::; ( 41f )-n/2e-1V1;coJ.

Then by Lebesgue's dominated convergence theorem and (8.40), we have

_lim V( Tie) = r _lim (' ( 41TTie)-nl^2 e-£('Yv(rie),rie) £ Jv (Tie) dμg(O) (V))

i->oo J~n i->oo.

= r (41Te)-n/2 _lim (Ti-n/2e-£i('Y..J7TV(0),0)dμg(rie))

JM= · i->oo

= r (41Te)-nl^2 e-e=(q,e) _lim dμ -r·(e)

J"Je_n i->oo g i

= f (41Te)-nl^2 e-e=(q,e)dμ 9 =(e) (q) = Voo(e).

JM=

Since V ( T) is a monotone decreasing function, we have

(8.51) V 00 (e) = 7->00 lim V(T) ~ V(oo).

In particular' v 00 ( e) is independent of e.