- REDUCED VOLUME FOR RICCI FLOW 393
Since n(p,0) (r1) = n (r1) ::::) n (r2) for r1 < r2, we have limT->0+ Xn(T) = 1,
where xn denotes the characteristic function of the set n. We computelim V (r) = lim r (47rr)-nf^2 e-£('yy(T),T) £ Jy (r) dx (V)
T->0+ T->0+ ln(T)= r lim [xn(T)(47rr)-nf^2 e-£('yy(T),T)_cJy(r)] dx(V)
lrpM T->O+= f 1. 7r-n/^2 e-IVl
2
dx (V) = 1,
}TpM
where we used (8.24).so(ii) From (8.22), we have for any 0 < r1 < r2 and VE D(r2),
r ;n/2 e-£('Yv(T 1 ),T 1 ) .C Jv ( ri) 2: r :;n/2 e-£('yy(T2),T2) .C Jv ( 72 ),
V (r2) = r (47rr2)-nf^2 e-£('Yy(T^2 )'^72 ) £Jy (r2) dx (V)
ln(T2)~ r (47rr1)-nf2e-£('Yv(T1),T1)_cJy(r1)dx(V)
ln(T2)~ { (47rr1)-n/2e-£('Yv(Ti),T1)_cJy(r1)dx(V)
ln(T1)
= V (r1),where we used n (r1) ::::) n (r2). Note that v (r) ~ 1 for any T > 0 follows
from (8.25) and (i).
(iii) We prove this statement in two steps by first showing that g(r)
is a shrinking gradient Ricci soliton and then showing that (M,g (r)) is
Euclidean space. If V ( r1) = V ( r2) for a pair of times 0 < r 1 < r 2 , then for
r E (r1,r2) and VE D(r), we have that equality in (8.22) holds:
:r [(47rr)-n/2 e-£('Yv(T),T) .CJv (r)J = 0,
which, by the proof of Lemma 8.16, implies that we have equality in (7.139).
Hence, by (7.140), we get
(8.26) Re h'v(r), r) +(Hess£) ('yv(r), r) =^9 h'v
2~), r)for all VE D(r) and r E [r1,r2], and where£ is 000 at ('yv(r),r) for all
such (V, r). Since V (r1) = V (r2), we have D (r1) = D (r2).
Suppose there exists Vi E TpM such that TVi ~ r1. Since n (r1) =!= 0,
there exists V2 E TpM such that rv 2 > r1. Since the function V f--t rv
is a continuous function, there exists 113 E TpM such that rv 3 E (r1, r2).
Thus 113 E D ( r1) - D ( r2) , which is a contradiction. Therefore, for every