1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

76 2. KAHLER-RICCI FLOW

4.2. The normalized Kahler-Ricci fl.ow equation. Let (Mn, J, go)

be a closed Kahler manifold. Now we make the basic assumption (corre-

sponding to the canonical case), holding for the rest of this section and

the next section, that the first Chern class is a real multiple of the Kahler

class, i.e., that

[po]= c [wo]

for some c E R Note that this is possible only if the first Chern class has

a sign, i.e., is negative definite, zero, or positive definite. Comparing (2.11)

and (2.12), we find that c = ~' where r ~ JM Rodμ 90 /Vol 90 (M) is the

average (complex) scalar curvature, so that

r 1

• 2
  
  
  • [wo] = -
    2
  
  
  
  

[po] = c1 (M).

7rn 7r

Sor depends only on the cohomology class [wo], n, and ci (M).

The normalized Kahler-Ricci flow is

(2.29)

The solution of (2.29) can be converted to the solution (2.26) by scaling the

metric and reparametrizing time, and vice versa (see Section 9 .1 in Chapter

6 of Volume One or subsection 9.1 below). Hence, from Lemma 2.36, we

know that the initial-value problem for (2.29) with g (0) =go has a solution

for a short time.

By a derivation similar to that of (2.28), we get the following para-

bolic (scalar) complex Monge-Ampere equation, corresponding to (2.29)

with gai3 (t) = g~i3 + 8a8(3cp (t),

8cp det (g~J + a,,88!.p) r

8t =log detgo_ + -;,,1P - Jo+ c1 (t)

"(8

(2.30)

for some function of time c1 (t). Here Jo is defined by Ra"r3 (go) - ~g~"r3 =

8aB(3Jo; this is possible because [-po+ ~wo] = 0.

4.3. Basic e~olution equations. Let g (t) be a solution of either the

Kahler-Ricci flow or the normalized Kahler-Ricci flow. We define the po-

tential function J = J (t) by

(2.31)

This equation is solvable since [-p + ~w J = 0 and by Lemma 2.26. Note

that J is determined up to an additive constant. Taking the trace of (2.31),

we have

(2.32) R-r = !:::.f.