1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

362 7. THE REDUCED DISTANCE

PROPOSITION 7.105 (Time-derivative of £-Jacobian). Let (Mn, g (r)),

r E [O, T] , be a solution of the backward Ricci flow with bounded sectional

curvature. Along a minimizing £-geodesic /'V (r), r E [O, rv ), with !'v(O) =

p, where rv is defined in Lemma 7.95, for 0 < f' < rv the £-Jacobian

£ J v ( r) satisfies

(7.139) ( -log£Jv d ) (r)::;---= n - -^1 3 K,

h ~ ~j

where K = K(!'v, r) is defined by (7.75). Equality in (7.139) holds at the

point !'v(r) only if

(7.140) Re (/'v(r), r) +(Hess£) (/'v(r), r) = g bv

2

~), r).

REMARK 7.106. The proof of (7.139) is closely modeled on that of the

classical Bishop-Gromov volume comparison theorem. Here we follow the

derivation using £-Jacobi fields. There are other ways to prove volume

comparison such as in Li [246].

REMARK 7.107. If we let Cf= 2./T and O" = 2Vi, then (7.139) says

(

dO" d log£ Jv ) (r) ::; - ?f^2 n

2 K + jj·

Compare this with (7.76).

PROOF OF PROPOSITION 7.105. Choose an orthonormal basis {Ei (r)}

of T"tv ('F) M. Since there is no point on /'V ( r), 0 ::; T ::; f', which is £-

conjugate to (p, 0) along /'v, we can extend Ei (r) to an £-Jacobi field Ei (r)

along /'V for r E [O, f] with Ei (0) = 0. Actually for the same reason, we

know that both {JY(r)} E T"tv(T)M and {Ei(r)} E T"tv(T)M are linearly

independent when r E (0, f]. We can write

n

JY (r) = 2:=A{Ej (.:r)

j=l

for some matrix (A{) EGL (n, ffi.). Then

n

JY (r) = 2:A{Ej (r)

j=l

for all r E [O, f], since we cannot have a nontrivial £-Jacobi field vanishing

at the endpoints r = 0, f. ·