1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

320 7. THE REDUCED DISTANCE

Plugging the above estimate into (7.70) and using IY (7)1^2 = ¥ IY (f)l^2 ,

we get

D

5.3. Laplacian comparison theorem for L. Tracing the Hessian

comparison theorem for L in (7.70), we obtain the Laplacian comparison

theorem for L. We adopt the notation in Corollary 7.40. Let {Ei}~ 1 be an

orthonormal basis at q = ry (f). For each i, we extend Ei to a vector field

Ei (7), 7 E [O, f], along ry to solve the ODE (7.66) with Ei (f) = Ei. Below,

Ei will stand for either Ei E TqM or Ei ( 7) , which will be clear from the

context. Taking Y = Ei in (7.70) and summing over i, we have

n

b.L (q, f) = L Hess(q,1') L (Ei, Ei)

i=l

n

• 2\l'fLRc(Ei,Ei) (f).
  i=l

We compute, without assuming (Ei (f) ,Ej (f)) = 5ij, that

d d

d7 (Ei, Ej) (7) = d7 [g (7) (Ei(7) 'Ej (7))]

= 2Rc(Ei,Ej) + (VxEi,Ej) + (Ei, VxEj)

= 2Rc(Ei,Ej) + (-Rc(Ei) + 2 ~Ei,Ej)

+ ( Ei, -Re (Ej) +

2

1

7

Ej)

1

= - (Ei,Ej) (7).

7

Hence

(7.72)

for all 7 E [O, f]. When i = j, we recover (7.68), which says IEi ( 7) 12 = 7 /f.