330 7. THE REDUCED DISTANCE6.4. The growth of the reduced distance function f. The growth
off, in particular the lower bound off, will be used to justify some technical
issues later (for example, the proof of Lemma 7.130). The next lemma
follows from Lemma 7.13(i) and (iii).LEMMA 7.59 (Bounds for the reduced distance). Let (Mn,g(r)), r E
[O, T] , be a complete solution to the backward Ricci flow with bounded sec-
tional curvature and let p be a basepoint. Then for any (q, f) EM x (0, T),
1 2 nCof _ e^200 T 2 nCof
4fe2CoTdg(O) (p, q) - -3-_:::; f, (q, r) _:::; 4f dg('f) (p, q) + -3-.Next we bound l\7£1^2 and I g; I by£, and hence we can bound them by
d;(T) (p, q). These estimates will be improved in Lemmas 7.64 and 7.65 when
we assume the curvature operator is nonnegative.LEMMA 7.60 (Bounds for first derivatives of£). Suppose (Mn,g(r)),
r E [O, T), is a complete solution to the backward Ricci flow with bounded
sectional curvature. Then for any f E (0, T) there exist positive constants
A 2: 1 and C1 depending only on f, n, T and Co which satisfy the following
properties. For any q EM and r E (0, f], we have f, (q, r) +Ar 2: 0,
(i)
2 C1
l\7£1 (q, r) :::; - (f (q, r) +Ar),
r
(ii)I ;: I ( q' r) :::; ~
1
( f, ( q' r) + Ar).PROOF. (i) Let 'Y(r) be a minimal £-geodesic from (p, 0) to (q, r) andlet X(i) ~ ~r By Lemma 7.13(ii) we have
l~X(r*)l^2 :::; n61f + f (q, f)
for some r* E (0, f). Here and below 61 and A are constants depending only
on f, n, T and Co and its value may change from line to line. Thus using
Shi's derivative estimate and integrating equation (7.45) (&-= 2Ji),
d 2 -2
d- lv'rx(r)I _ =-&-Re (v'rx(r), v'rx(r)) + <!.__ lvR, v'fx(r))
O" g(u^2 /4) 4 \
on [r, r*], as in the proof of Lemma 7.28, we get for any r E [O, f]
IJTX(r)l^2 :::; ( n6 1 f + f (q, f) + 61 f) e^61.
It follows from 'Vf(q, r) = X(r) that
(7.98) [\7£[^2 (q,r)::=;--:;-^61 ( f(q,r)+Ar. A )
In particular f (q, r) +Ar 2: 0 for all (q, r).