1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

10. LINEAR AND INTERPOLATED DIFFERENTIAL HARNACK ESTIMATES 123

PROOF. First we observe that the equivalence of (2.140) and (2.141)

follows from the fact that the minimizing (1, 0)-form Va for the LHS of (2.140)

is equal to -1;; and dividing (2.140) by u. We compute

8f 2

at = Lif + IV' ,,,JI + c:R.

Using (2.22) and commuting a pair of derivatives, we have

:tj ai3 = \7 a \7 i3 ( Lij + l\7 ,,,Jl

2

+ER)

(2.142) = LiLfai] + E\7 a \7 i]R + fa,,,fi];y + J a;yfi]')'

+ \7;yf\7')'fai] + V',,,JV';yfai] + Ra;y8i]\75f\7'1'j.

Using the analogue of (2.105) for the c:-speed Kahler-Ricci fl.ow,

we then compute

:t (fai] + ERai] + ~9ai3) = LiL Uai3 + ERai]) + f a')'fi];y + fa;yfi]'I'

+ \7;yf\7'/'fai] + V',,,JV';yfai] + Ra;y8i]\75J\7,,,J

+ E^2 LiRai3 + E^2 Rai3'1'8R8;y - E

2

Ra;yR,,,i]

1 E

• t2 9ai3 - t Rai]·

Hence letting Saf3 ~ f <:<i3 + c:Rai3 + t9ai3 and using (2.106), we have

( :t - LiL) sai3 = fa,,,fi];y + E

2

( LiRai3 + Rai],,,8R8;y + :t Rai3)

• EV';yf\7 ')'Rai] - E\7 ')'f\7 ;yRai] + Ra;y8i]\75f\7 'l'J

• \7;yf\7'/'Sai3 + V',,,f\7;ySai3

+ ~Sa;y (ii],,, - c:R,,,i] - ~9,,,i])

• t (fa;y - ERa;y - ~9a;y) Si],,,·

Now for the c:-speed Kahler-Ricci fl.ow, by Cao's Kahler matrix differ-

ential Harnack estimate (2.111) with Xa = -:\7af, we have

1

0 :S LiRai3 + Rai]')'8R8;y +Et Rai]

1 1

• -\7;yf\7'/'Rai3 - -\7,,,J\7;yRai3
  E E
  1

• 2Ra;y8i3\75f\7 E ,,,J.