1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

114 2. KAHLER-RICCI FLOW

9.3. Proof of the matrix Harnack estimate. In this subsection we

prove Theorem 2.87. Let

The following computation is the one corresponding to Proposition 2.79.

PROPOSITION 2.93 (Evolution of the Harnack quantity Zcx13). Suppose

that a vector field X satisfies

-- 1

V'JXy = V' oX;y = R'YJ + t9ryJ,

Y'oXry = V'JX;y = 0,

and

Then Zcxi3 = Z (X)cxi3 defined by (2.111) satisfies the evolution equation:

(2.119)

(:t -~) Zcxi3 = Raf3ryJZ-yo - ~ (Ra1Z'Y13 + R'Y13Za1)

2

+ PcxJryP1513;y - Pcx7JPi3'Yo - tzcxi3·

The proposition follows from Lemmas 2.94 and 2.95 below.
Assuming the proposition, we now prove the differential Harnack esti-
mate, i.e., Proposition 2.93. Applying Zcxi3 to a (1, 0)-vector field W, we have
the following general formula:

(! -~) ( Zcx13wcxwi3) = ( (! -~) Zcxi3) wcxwi3

• 
  

  zcxi3 ( (:t -~) (wcxwi§))

  
  


• 
  

  V' rZ cxi3 V' T ( wcxwi3) + V' :rZ cxi3 V' r ( wcxwi3) '

  
  

where wiJ ~ W.B. Thus, if we have a null vector W of Zcxi3 at a point (xo, to)
and if we extend W locally in space and time so that at (x 0 , t 0 )

(2.120)

(2.121)