414 8. APPLICATIONS OF THE REDUCED DISTANCE
(ii) Note that V(oo) < 1 follows from Corollary 8.17(iii). To see V(oo) >
0, we compute using (8.46) and e = 1 that
Vh) = f (47r)-n;2 e-ei(q,1)dμ9r·(1)(q)
JM i2'.:: 1 (47r)-n/2 e-ei(q,l)dμ9r·(l)(q)
B9r· i (1) (qri ,c:-1/2) i2::: (47rA)-n/^2 e~^8 -1
Vol 9 ri(l) B 9 ri(l) (qTi,c-l/^2 )= ( 47r A)-n/2 e _^0 -1 ·Ti -n/2 u vOg(Ti) l B 9h) ( qTi,Ti 1/2 c -1/2).
By Lemma 8.35, we have R(q,Ti) ::::;^8 ~
1
on the ball B 9 ( 7 i)(q 7 i,Tl^12 c-^1 /^2 ).
It follows from g(T) being K-noncollapsed on all scales (choosing the scale
r = min { Tl/^2 c-^1 /^2 , Tl/^2 0112 }) that
Ti -n/2 u V09(Ti)^1 B 9h) ( qTi,Ti 1/2 c -1/2) 2'.::K· ( mm · { c -1/2 ,u d/2})n.
Hence
Vb) 2::: (47rA)-n/2e-o-1r;;. (min{c-1/2,51/2})n
and V(oo) > 0.
(iii) For any 1/J(B) which has compact support in (A-^1 ,A), we compute{A { (47re)-n/2 e-f!oo(q,B)'l/J'(())dμ9oo(B)(q)d()
}A-1 JM 00= {A V 00 (())'ljJ'(())d() = V 00 (0) {A 1/J'(O)d() = 0.
}A-1 JA-1
D4.5. The limit is a shrinking gradient soliton. Let 1/J(O) 2::: 0 bea smooth function only of () with compact support in (A -1, A). Applying
Stokes's theorem for Lipschitz functions, we get
{A r (47re)-% (&foo -R + !!_) e-f!oo(q,B)'l/J (0) dμ (B) (q) d()
}A-1}Moo {)() 900 2() 900.= h~1 JM (47r0)-% e-e=(q,B)'l/J' (0) dμ9oo(B) (q) d()