- EQUATIONS AND INEQUALITIES SATISFIED BYLAND .e 333
where C = 1 + ~· If T = oo, then for any (q, f) EM x (0, oo)
. 3.
1vc1^2 (q, r) + R (q, r) ::::; -=c (q, r).
T
Hence, for an ancient solution with bounded Rm 2: 0, the reduced dis-
tance bounds Rm.
Now we can estimate I g~ I·
LEMMA 7.65. Let (Mn, g ( T)) , T E [O, T] , be a solution to the backward
Ricci flow with bounded nonnegative curvature operator. If f E (0, (1 - c) T)
for some c E (0, 1), then at any (q, f) EM x· (0, oo),
I
oc/ < c+ 1 ~
OT - 2 f'
where C = 1 + ~· If T = oo, then at any (q, f) EM x (0, oo),
(7.100) 1~logc/ OT < - 3. f'
and for any 0 <Ti< T2 and q EM,
(7.101) (T1)
2
::::; C(q,T2) ::::; (T2)2
T2 C(q,T1) Ti
PROOF. From (7.94) we have at (q, f) thatoc + ~ 1vc1^2 = ~R--~--e.
OT 2 2 2fHence by Lemma 7.64 and£ (x, T) > 0, we have
/;:/::::; ~ (1vc12
+ R) + 2 ~£<C+lc.
- 2f
If T = oo, we can choose c arbitrarily close to 1 and get I g 7 log £1 ::::; ~·
Since log C(x, T) is a locally Lipschitz function of T > 0, we have
/log C(q,T2) I::::; 172 3.dT =log (T2)2'
C(q,T1) 71 T Tland (7.101) follows. D
6.6. Reduced distance under Cheeger-Gromov convergence.
Finally we discuss the convergence of the reduced distance under Cheeger-
Gromov convergence. Let. {(Mk, 9k (T) ,pk)}kEN and (M~, 900 (T) ,Poo),
T E [O, T], be complete pointed solutions to the backward Ricci flow satisfy-
ing the curvature bound
rriax {1Rm 9 k I, IRc 9 1< I} ::::; Co < oo on Mk x [O, T]
for all k E N. Suppose that (Mk,9k (T) ,pk) ---+ (M 00 ,g 00 (T) ,p 00 ) on the
time interval [O, T] in the C^00 Cheeger-Gromov sense; that is, there exist