86 2. KAHLER-RICCI FLOW(using the Kahler identities (2.2) to get the second equality in (2.67)). Note
thatWe claim that the Cauchy-Schwarz inequality gives(2.68) j\7YJ2:::; y (g (ot.6 g'Y8gap,8ga,6 og.,,,!!).
oz'Y f)z8
By applying (2.68) to (2.66), we then obtain (2.65), as desired.
To prove (2.68), we first observe that we may further assume
9a,B(x, t) = Aa5a(3
is diagonal, where A.a is defined above. The LHS of (2.68) can be written at
x as<LA. ·L:~l 18gaa/
2- u u Ol,'Y AaA'Y oz'Y
<" - ~Au. " ~ AaA'Y 1 I 09a,612 oz'Y
u a,f3,'Y= y ( 9 (O)v,6 9 'Y'S 9 ap,^0 9a,6 8g.,,,!!).
f)z'Y f)z8
Thus the claimed inequality (2.68) and the lemma are both proved. D
Next we prove the key C^2 -estimate (2.62) via an application of the max-
imum principle to (2.63).
PROPOSITION 2.59 (C^2 -estimate for c.p). Let c.p (t), t E [0, T), where
T E (0, oo], be a solution of the NKRF (2.50) on a closed Kahler manifold