1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

9. MATRIX DIFFERENTIAL HARNACK ESTIMATE 111

Before we prove the corollary, we first recall how to go from the Kahler-

Ricci flow to the normalized Kahler-Ricci flow on closed manifolds. Let g (t)

be a solution to gt9a(3 = -Ra/3 and let g (t) ~ 'ljJ (t) g (t), where 'ljJ (t) is to

be defined below. We compute

{) -.

{)tga/3 = -'1/JRa/3 + '1/Jga/3

since Ra/3 = Ra/3. Hence if we define a new time parameter t by %t = ~ gt,

we have

{)_ - ~-

{)t9a(3 = -Ra/3 + '1/J29a(3·

In particular, to obtain the normalized Kahler-Ricci flow, where g (t) re-

mains in the same Kahler class, we set

~ (t) r

'lj;(t)^2 n'

where r is the average scalar curvature of g (t), which is independent of

time since g (t) stays in the same Kahler class. Thus we take 'ljJ (t) ~

(1-~t)-

1

. Since dt = 'lj;dt, we may take t ~ -¥log (1-~t). That is,

t = ¥ ( 1 - e-F).

PROOF OF COROLLARY 2.90. Let g (i) be a solution of the normalized

Kahler-Ricci flow %r9a(3 = -Ra/3 + ~9a(3· Then

is a solution of the Kahler-Ricci flow and we have the estimate ~~~~'.;~j >

~e-tA. This implies

where

R (x2, f2)

R (x1, f1)

LS.(x1,t1;x2,t2) =inflt

2

ldd'Y (t)l

2

dt=inf ~£

2

ldr_ (i)l

2

_ dt

'Y ti t g(t) 'Y 1£1 dt g(t)

since it = '1/J"it, g (t) = 'ljJ-^19 (i), and dt = 'ljJ-^1 di. D

Since g (t) has nonnegative bisectional curvature, and in particular it
has nonnegative Ricci curvature, under the normalized Kahler-Ricci flow

we have gt9a(3::::; ~9a(3' which implies g (t)::::; e;;;(t-ti)g (t1) fort :2': ti. Hence

(2.115) implies