1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. FIRST VARIATION OF C-LENGTH AND EXISTENCE OF C-GEODESICS 303

{3° - (0) = 0 and d~o dri" (er) =f. 0 for er > 0, then {3° - (er) = Aer^2 for some positive

constant A.

SOLUTION TO EXERCISE 7.23. If p^0 (er)= 7 (er)' then the time compo-

nent of the geodesic equation with respect to '\7 is

d2 po """" (-o -) dpi dPJ

0 = der2 + ~ rij 0 f3 -d -d

O< _i,J_n .. < er er

d

2

7 1 (d7)

2

= der^2 - 27 der

since I'?j = 0 when i 2: 1 or j 2: 1, and rgo

7 (er) > 0 and ~:(er) > 0 for er> 0, we have

• 2 ~. Hence, assuming

so that

d d7 B dr d

-log-= du =du= -logy'T

der der ~: 27 der '

d7 = Cy!T
der
for some constant C > 0. Since 7 (O).= 0, we conclude

er2

7 (er)= C^2 4.

3.4. Existence of £-geodesics. Our next order of business is to estab-

lish the existence of solutions to the initial-value problem for the £-geodesic
equation. In this subsection (Mn, g ( 7)) , 7 E [O, T] , is a complete solution to
the backward Ricci fl.ow with curvature bound max {IRml, !Rel} S Co < oo
on M x [O, T]. We shall use the following

LEMMA 7.24 (Estimate for speed of £-geodesics). Let (Mn,g(7)), 7 E

[O, T], be a solution to the backward Ricci flow with bounded sectional cur-

vature. There exists a constant C ( n) < oo depending only on n such that

given 0 S 71 S 72 < T, if/ : h, 72] -+ M is an £-geodesic with

lim y'Tdd/ (7) =VE T'Y(ri)M,

r-->r1 7

then for any 7 E [Ti, 72],

7 I d1 (7)12 S e6GoT IVl2 +. C(n)T -1 (e6CoT - 1)'

d7 g(r) mm {T-72, C 0 }

where Co is as in (7.27) and IVl^2 ~ IVl!(ri) ·

PROOF. Let eri ~ 2..Jii. Define f3: [er1,er2]-+ M by f3(er) = 1(er^2 /4),

so that

(7.43) lim df3 = 1Vl^2.

I 1

2

u-->u1 der g( u2 I 4 )