- EXISTENCE OF AN ASYMPTOTIC SHRINKER in
Thus2(n-1) (rt 7 ) +K(7)r(7))
:::::: J6(n+1) 7i/47-3/4 ( ve (q2, 7) + ~7-i/47i/4).
By applying this to (19.53), we conclude from (19.51) thatd 9 (r) (qi, qz) S v'6(n+1) J,' r^1 /^4 7-^3 /^4 ( ~ (q 2 , f) + ~ r-^1 /^471 /^4 ) d7
+ 4J37i/2 ( gi/2 (qi, 7) + gi/2 (q2, 7))Therefore= 4J6 (n + 1) 7i/^2 .Je (q2, 7) + 33 /^42 (n + 1) 7i/^2
+ 4J37i/2 (ei/2 (qi, 7) + gi/2 (q2, 7)):::;: 4J6 (n + 2) 7i/2 (ei/2 (qi, 7) + gi/2 (q2, 7))
+ 33 /^4 2(n+1) 7i/^2.
gi/2 ( 7 ) + gi/2 ( 7 ) > 1 dg(r) (qi, q2)
qi, q2, - 4v'6 (n + 2) 7i/2since^3114 2(n+i) i^8
4 V6(n+ 2 ) :::;:^2. We finally obtain (19.46) from2
1
24 ( 1)
2
e (qi, 7) + e (q2, 7) + 1::::::
9gi/^2 (qi, 7) + gi/^2 (q2, 7) +
25.1.3. Estimates for f and R in large neighborhoods of (qn 7).
The following is Lemma 8.35 in Part I.DLEMMA 19.47. Let (Mn, g (t))) t :::;: 0) be a K,-solution with basepoint(p, 0). Let f (q, 7) be the reduced distance of g (-7) and for each 7 > 0 let qr
(^28) For any a, b > 0 we have
a^2 + b^2 + 1 > -4 ( a+ b + -1)
2
- 9 2
since
a^2 + b^2 + 1 - c (a + b + ~)
2= (1 - c) a^2 + (1 - c) b^2 - 2cab - ca - cb + ( 1 - D
> (1 - 2c) a^2 - ca+ -1 ( 1 - -c) + (1 - 2c) b^2 - cb + -1 ( 1 - -c).
- 2 4 2 4