324 7. THE REDUCED DISTANCE
that is,
(7.85) (:r +il) (L-2nr) :S 0.
Now we prove a lemma which describes the limiting behavior of L (q, 7)
as 7-+ O+. We will use it to·estimate the minimal value of 2 ./¥L(·,7) (and
likewise L) over M..
LEMMA 7.47 (L tends to its Euclidean value as 7-+ 0). We have
(7.86) _lim L(q,7) = (dg(O) (p,q))
2
.
T-+0+Hence L satisfies
lim L (q,^7 ) = 1.
7-+0+ [dg(O) (p, q)]2 I 2y'f
PROOF. We only need to prove the first equality above; the second
equality follows by definition. In Lemma 7.13(i), let r = r2 = 7 and let
'Y : [O, 7] -+ M be a minimal £-geodesic between p and q. We have
d;(o)(P, q) :S e2Co7 ( L(q, 7) + 4n3Co 72)
for any 7 E (0, T). Taking the limit lim 7 ...... o+ of the above inequality, we get
lim7 ...... o+ L ( q, 7) 2: ( d 9 (o) (p, q))
2
.
To see the other direction of the inequality, choosing r2 = 7 in Lemma
7.13(iii), we have
L(q - 7) < --7 4nCo^2 + e 2c 0 - 2^7 d _ (p q)
' - 3 g(T) )for any 7 E (0, T). Taking 7-+ O+, we get
_lim L (q, 7) :S d;(o) (p, q).
T-+0+The lemma is proved. D
Next we estimate the minimum value of L (q, 7) - 2n7 over M.
LEMMA 7.48 (Monotonicity and estimate for minM L). Let (Mn,g (r)),r E [O, T], be a complete solution to the backward Ricci ftow with bounded
sectional curvature,
(i) The functionmin (L (q, f) - 2n7)
qEM
is a nonincreasing function of 7.
(ii) For all 7 E (0, T) ,min L (q, 7) :S 2nf.
qEM