1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

8. .C-JACOBI FIELDS AND THE .C-EXPONENTIAL MAP 351

By (7.69), since (7.126) holds and Y (0) = 0, we have

(oic) ('Y) - 2v1f ( x, \7yY) (r)

= -1

7

y!TH ( X, Y) d7 + IY $)1

2

- 2vfrRc (Y, Y) (f).

Note that is (f) =rs (r) implies \7 yY (r) = \7y Y (r). Hence

LEMMA 7.87 (Differential inequality for length of £-Jacobi field). Let

/s : [O, 72] -t M, where 72 E (0, T), s E (-c, c), and c > 0, be a smooth

family of minimal £-geodesics with Ys (0) = 0 for s E (-c, c). Then for any

f E. (0, 72] the £-Jacobi field Y ( 7) = dJ; ls=O satisfies the estimate

(1.129) .:!:_I 1Yl^2 s -~ r7 v!TH ( x, f) d7 + IY ~)1

2

,

d7 T=T .jT lo 7

where Y satisfies (7.126) and (7.127) and H ( X, Y) is Hamilton's Harnack

quantity defined in (7.63).

Note that the only place where we used an inequality (versus an equality)

in our derivation is (8-f.£) (1) S (of£) ('Y). Hence equality holds in (7.129)

if and only if the vector field Y satisfying (7.126) and (7.127) is an £-Jacobi
field. Then

d~IT=T IYl

2

= d~IT=T lyl

2

= IY~)l

2

in (7.129), and J; y!TH ( X, Y) d7 = 0. From (7.125) we get

(7.130) ..:!:__I IYl^2 = 2 Re (Y, Y) (f) + ~ (Hess L) (Y, Y) (f) = IY ~) 1

2

b~ y7 7

Applying Hamilton's matrix Harnack inequality to (7.129), we get

LEMMA 7.88 (Estimate for time-derivative of length of £-Jacobi field). If

the solution (Mn, g ( 7)) , 7 E [O, T] , to the backward Ricci flow has bounded

nonnegative curvature operator and the £-Jacobi field Y ( 7) along a minimal

£-geodesic / : [O, 72] -t M satisfies Y (0) = 0, then for c E (0, 1) and

f E (0, min { 72, (1 - c)T}],

..:!:__I log IYl^2 S ~ (GR(1 (f), f) + 1),

d7 T=T 7

where G = ~· If T = oo, then

- di loglYl^2 S ::(2£(1(f),f)+l).^1

d7 T=T 7