1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. EXISTENCE OF AN ASYMPTOTIC SHRINKER 115

STEP 3. Limit of the reduced volumes.

The reduced volume (with basepoint po) of the solution 9ri (0) is

VgTi (0) ~ r (47rO)-n/^2 e_.egTi(q,fJ)dμ9r·(fJ) (q) for 0 E (0, oo).

}M i

Corresponding to the limit solution (M~,g 00 (0),q 00 ) in (19.60), define the
mock reduced volume by

(19.62) Voo(O) ~ r (47rO)-nl^2 e-.eoo(q,fJ)dμgoo(e)(q) for 0 E (0, oo)'

}Moo

where £ 00 is given by (19.61). The aforementioned issue we neglected to
discuss (on p. 413 of Part I) is in regards to showing that V 00 ( 0) is finite and

(19.63)

The rest of this discussion is devoted to proving this equality. Fix

any 0 E (O,oo), c > O, and consider the balls B 9 ri(fJ) (qri' ~) C M and

Bg 00 (fJ) (qoo, ~) C Moo. Clearly

1 ( 1) (41f0)-n/2 e-.e9Ti(q,fJ)dμ9ri(fJ) (q)

Bgri(IJ) Qri'e

= r ( -1( ) 1) (47r0)-n/2 e-.ei(q,fJ)dμ9i(fJ) (q)

j Bgi (B) <I\ Qri , e

(19.64) --+ 1 1 (47rO)-n/^2 e-.eoo(q,fJ)dμgoo(fJ)(q)

Bgoo(B) (qoo,e)

(note that <I>i (qoo) = qrJ·

Now we recall Perelman's estimate for the reduced distance. By (19.46)

we have that there exists a constant c ( n) > 0 depending only on n such

that for any q E M and 0 E (0, oo)

(19.65)

· c (n) d^2 (fJ) (q, qrJ

-f9ri (q, 0) :S f9ri (qri, 0) + 1 - 9rio

(19.66)

< n+2 - c(n) d;ri(fJ)(q,qrJ.

• 2 o

This is the main estimate we shall use to prove (19.63).
Pulling back by i and taking the limit, we have