1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. EXISTENCE AND CONVERGENCE 85

constant C' < oo. Thus a bound on Y shall imply an estimate of the complex

Hessian of <p, i.e.,

(2.62)

for some C' < oo. By abuse of notation, we shall call this the C^2 -estimate.

REMARK 2.57. A quantity similar to Y also played an important role in

the later work of Donaldson [128] on the Hermitian-Einstein flow.

We now turn to estimating Y. First we apply the heat operator to log Y.

LEMMA 2.58. We have

(

0 ) g'Y

(^8) (t)gai3(t)Ra{(o) r ·
(2.63) -at - ~ log Y < -- y 'Y. + -n'
where~~ ~g(t) and R~/ (0) ~ gi3TJ (0) R~ 8 TJ(O).
PROOF. From (2.59) and (2.29), we compute
(2.64) m~ ~^1 y = Y! 9 ai3(o)~ m (^9) ~ -(t) = -gai3(o)Rai3(t) Y + ~y
Thus, to prove the lemma, it suffices to show that
(2.65) Y ~logY 2:: g'Y^8 (t)gai3(t)R~{(o) - gai3(o)Rai3(t).

Given any point x E M, we will calculate in a local holomorphic coordi-

nate system which is normal with respect to the metric g(O) at x, so that
a~a9(3;y (x, 0) = 0 and 9ai3 (x, 0) = Oaf3· To simplify notation, we adopt the
convention that the quantities below are at time t, unless there is a (0) after
them, in which case they are at time 0. Since from (2.4), at x,

• 32 32

R~8μ, (O) = -gaf3 (O) 8z'Yaz 09 μ,i3 (O) = -8z'Y8z 89 μ,°' (O)'

we compute that the Laplacian of Y at x is given by

~y = g'Y8 32 (gai3(0)gai3)

8z'Yaz^8

• ry8 - 32 - ( ) ry8 ai3 ( ) 32 -


• g 9af3 8z'Yazogf3a 0 + g g 0 8z'Yaz8gaf3
  
  
  • 
    
    
    
    • 
      
      
      
      • 32
        = g'Y8 g f3-Ra j3 (0) + gaf3 (O)g"f8 g f3-
        a ry8 8z'Yaz8 a
      
      
      
      
    
    
    
    
  
  
  
  

(2.66) = g'Y^8 R~{(O)gai3 - gai3(o)Rai3

ai3 ( ) o;y A'fi 8 8 -

+ g 0 g g az8 9ar; az'Y 9>.(3'

where we. used (2.5) and

(2.67)