1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

408 8. APPLICATIONS OF THE REDUCED DISTANCE

PROOF. Given that we know how curvature changes under scaling, it is

easy to check that 'Yi (re) satisfies the £-geodesic equation for the solution

g,,. ( e). We compute using the change of variable 7 = re that

lim yed"fi (re) = lim Vfd1i_(r) VT= ../TV

e--+O+ de 1'--+0+ dr

By the uniqueness of the initial-value problem for the £-geodesic equation,

we get (8.40).

Given any curve a (i), i E [O, re], from (po, 0) to (q, re), the curve

a,,. (e) ~ 'Y (re) , 0 E [O, e], joins (po, 0) to (q, e). We compute

~£9 (a)= ~ re v'r (R9('f) +I dd~ 12 ) di

2y re 2y re lo r 9 (7)

= ~ re Vre (r-1R (e) +r-1 lda:12 -) rdB

2vrelo^9 r de 9 r(e)

• _1 re JO (R - + Ida,,. 12 ) dB

- 2..Je lo^9 r(e) dB 9 r(e)

1

= 2ve£9r (a,,.).

Since £9 (q,re) = infa [ 2 )Te£9 (a)] and f,9r (q,e) = inf/3 [ 2 ~£9r (,B)], we

conclude that f,9 ( q' re) = f,9r ( q' e) and the minimizing £-geodesics are re-

lated by (8.40).

From (8.40) we get

£~e exp : (Tp 0 M, g (0, Po)) --+ (M, g (re)) where V --+'Yi (re),

t,~r exp: (Tp 0 M, r-^1 g (O,po)) --+ (M, r-^1 g (re)) where ../TV--+ 'Yi (re).

Hence the Jacobian of the above two maps are related by (8.40). D

4.2. The blowdown limit of g,,.i(e). Recall by Lemma 7.64 that the

estimate

(8.43) [\7£ (q, r)J2 + R (q, r) :S 3£ (q, r)
r
holds for the solution g ( r). We have the following consequence in regards
to the space-time points ( q,,., r).

LEMMA 8.35 (Estimates for f, and R in large neighborhoods of ( q,,., r)).

For any E > 0 and A > 1, there exists 8 > 0 such that for any r > 0,

£(q, i) :S 5-l and iR (q,T) :S 5-^1

for all (q,i) E B 9 (r) (q,,., ~) x [A-^1 r,Ar].