1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

90 2. KAHLER-RICCI FLOW

simplicity we drop the indices k in our notation below. Since g is Kahler, we

may write 9a/J locally as the complex (Hermitian) Hessian ua/J of a function

u. We shall show that the second derivative of u has bounded Holder norm.

Now the equation reads locally as

(2.80) log det ( UafJ) = h.

It is convenient to write log det as a function F(p), where plies in the domain

of positive definite Hermitian symmetric matrices. An important property

that we shall make full use of is that F(p) = log det(p) is a concave function

of p, a fact which can be easily checked. Taking the derivative of (2.80), we

have

8F

-a -ua/J'Y = h'Y,

Pa.(3

82 F 8F

8 -8 -UafJ'YUμJYy + -8 -UaiJ'Y'? = h'Y;y

Pa.(3 Pμv Pa.(3

for each 'Y = 1, ... , n. By the concavity of F we have

8F

(2.81) ~Ua/J'Yi?: h'Y'?·

UPa.(3

On the other hand recall that

8F af3- af3-

--= u =g '

8pa/J

where ( uaiJ) is the inverse of ( ua/J) = (9a(J).^5 Therefore we can rewrite
(2.81) as
(2.82) /:':;.uw ?: h'Y;y,
where
W ~ U'Y'Y
and /:':;.u denotes the Laplacian with respect to the metric 9a/J·
For any R::::; Ro, let
(2.83) M(s) = sup w and m(s) = inf w.
B(sR) B(sR)
Also define the oscillation function:
w(sR) ~ M(s) - m(s).
The following weak form of the Harnack inequality, which holds in gen-
eral for linear elliptic operators of divergence form, plays a crucial role in
our estimate. One can find the proof of this result in various papers and

(^5) This is not different from the real version of this formula used in Volume One, which
is the variation formula

-^8 log <let A-· = ( A -l)ij -A-^8 ·
& ~ & ~
for any invertible matrix Aij.