- EXISTENCE AND CONVERGENCE 87
(1) (Uniform estimate when c1 ::::; 0) If c1 (M) ::::; 0, then there exists a
constant C < oo depending only on the initial metric such that
(2.69) Y(x, t) ::::; C
for all (x, t) EM x [O, T). Hence there exists C' < oo, independent
of T, such that the complex Hessian of <p satisfies(2.70) IVa\7J3<p(t)l 9 (0)::::; C'on M x [O,T).(2) (Time-dependent estimate when ci > 0) If c 1 (M) > 0, then there
exist constants C and C', both depending on g (0) and T < oo, such
that the above estimates (2.69) and (2.70) hold on M x [O, T).REMARK 2.60. Estimate (2.70) implies l.6. 9 (o)'PI ::::; C'. On the otherhand, by Lemma 2.53, we have l'PI ::::; Co for some Co < oo. Hence, by
standard elliptic theory, we have for any a E (0, 1), ll'Pllo1,"' ::::; C1 for some
C1 < oo depending on a. However at this stage of the proof, it is not clear
whether !Hess 'Pl is bounded, where Hess denotes the real Hessian, but we
do not need this.
PROOF. By the discussion above, regarding inequality (2.62), we only
need to bound Y from above. Again, we calculate in local holomorphic co-
ordinates around x where 9aJ3(x, 0) = 6a13, 0 ~ 7 gai3(x, 0) = 0, and 9aJ3(x, t) =
.a5a(3· By (2.63) and Aa ::::; Y, we have
(2.71) (~ - .6.) logY < -Racr(y (0) Aa + !:._ < C1 t .!_ + !:.
at - y >-1 n - 1=1 >-1 n'
where C 1 is a constant depending only on a lower bound of the bisectional
curvatures of g(O). To control the bad terms on the RHS above, we consider
equation (2.54) ·
(2.72) (-a -.6. ) <p = -f - .6.<p = - f -n + I: n - ,^1
m ~1~where the second equality follows from (2.60). Consider the modified quan-
tity
w = logY - (C1 + l)<p.
Combining (2.71) and (2.72), we have
(
a ) n 1
--.6. at w<-'°"""-+G2 - L.J>. '
1=1 1(2.73)where C2 depends on C1 and llflloo (if r ::::; 0, then C2 is independent of
time, and if r > 0, then C 2 depends on time). By the maximum principle,
(2. 73) implies that w can be bounded above on M x [O, T) by a constant C
depending on T.