1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

314 7. THE REDUCED DISTANCE

On the other hand, we compute

d

dT (\lyY,X) = (\lx\lyY,X) + (\lyY, \lxX)

• ~~ (\1 y Y, X) + ( ( :T \l) y Y, X).

Now~~= 2Rc and

( ( : 7 \l) y Y, X) = 2 (Vy Re) (Y, X) - (\1 x Re) (Y, Y).

Hence

(7.60)

d

dT (\lyY,X) = (\lx\lyY, X) + (\lyY, \lxX) + 2Rc (\lyY, X)

+ 2 (\ly Re) (Y, X) - (\1 x Re) (Y, Y).

Suppose
(7.61) y (0) = 0
(this and the fact that ft X ( T) has a limit as T ---+ 0 are used to get the
third equality below). Then applying (7.60) to (7.59) and integrating by
parts, we compute

( 8f C) ('Y)

= 1

7

VT (Y (Y (R)) + 2 (R(Y, X) Y, X) + 2 l\lyX1^2 ) dT

+2 yT ~

1

(^7) r::: ( d~ (\ly Y, X) - (\ly Y, \l xX) - 2 Re (\ly Y, X) )
o -2 (\ly Re) (Y, X) + (\1 x Re) (Y, Y)
= 1
7

VT (Y (Y (R)) + 2 (R (Y, X) Y, X) + 2 J\lyXJ^2 ) dT

+ 2 yT dT

1

(^7) r::: ( - (\ly Y, \l xX) - 2 Re (\ly Y, X) )
o -2 (\ly Re) (Y, X) + (\1 x Re) (Y, Y)

• 2yfi (V'y Y, X) I~ -fo'f ~ (\ly Y, X) dT

= 2yT (\lyY. X) + Vi dT

~ 1'f (Y(Y(R))-\lyY·\lR )

' o +2 (R(Y, X) Y, X) + 2 J\lyXJ^2

+ 2 yT r dT

1

'f r:::(-(\lyY,[\lxX+2Re(X)-~\lR+2^1 X]))

o -2 (\ly Re) (Y, X) + (\1 x Re) (Y, Y)

= 2./f-(\lyY, X) + 1

7

VT (v},yR + 2 (R (Y, X) Y, X) + 2 J\lyXJ^2 ) dT

• 1

7

VT (-4 (\ly Re) (Y, X) + 2 (V x Re) (Y, Y)) dT,