1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

88 2. KAHLER-RICCI FLOW

To get a bound for w independent of T when r ::; 0, we need to work

harder. When r ::; 0, we shall use the term - L:'Y ;..^17 to dominate (from

below) a function of w which approaches -oo as w -+ oo. Using equation

(2.50), f = -fftcp, and (2.61), we have

r det 9a,a(x, t)

f (x, t) - f (x, 0) + -cp(x, t) = - log d t -( O),

n e 9af3 x,

and hence

y ef(x,t)-f(x,O)+f;,cp(x,t) = L Aa. II:

a 'Y 'Y

= L (rr 2-) < (~ :a)n-l

a ')"/=a A')' ~

using a standard inequality (we dropped a factor of nn^1 _ 2 since it makes the

inequality easier to see). Since f and cp are uniformly bounded, this implies

that there exists a constant C3, which only depends on the initial data, so

that

(2.74)

Notice that ew = e-(Ci +l)cpy: We then have

(2.75) ( )

n-1

ew ::; C4 ~ Ala '

where C4 > 0 only depends on the initial data. Combining (2.73) and (2.75),

we have

(2.76) ( -^8 - 6. ) w < --en-l^1 w + C2.

8t - C4

This implies that, at any time t where Wmax (t) ~ (n-1) log (C2C4), we

have ftwmax (t) ::; 0. Hence, by the maximum principle,

sup w (x, t) ::; max {sup w (x, 0), (n - 1) log ( C2C4)}

xEJvi xEJvi

for all t E [O, T), and hence Y is also bounded from above. When r::; 0, both

the constant C 2 and the function cp are uniformly bounded independent of

T; hence Y is uniformly bounded independent of T. D

Now we can complete the proof of Theorem 2.50.

PROOF OF THEOREM 2.50. Let cp (t), t E [0, T), be a solution of t~e

NKRF (2.50) on a closed Kahler manifold (Mn, go), where Tis the maximal

time of existence. If T < oo, then by the C^2 -estimate (i.e., the estimate for
the complex Hessian of cp), the metrics g(t) are uniformly equivalent to g(O)