1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. EXISTENCE AND CONVERGENCE 81

5.1. Cao's existence and convergence theorem. When the first

Chern class has a definite sign, either negative, zero, or positive, Cao proved

that the normalized Kahler-Ricci flow exists for all time. Let KRF and

NKRF denote the Kahler-Ricci flow and the normalized Kahler-Ricci flow,

respectively.

THEOREM 2.50 (NKRF: ci <, =, > 0 global existence). Let (Mn, go) be

a closed Kahler manifold with

(1) either c1 (M) < 0, ci (M) = 0, or c1 (M) > 0, and

(2) r;: [wo] = 27rc1 (M).

Then there exists a unique solution g(t) of the normalized Kahler-Ricci flow

defined for all t E [O, oo) with g(O) =go.

When the first Chern class is nonpositive, Cao proved that the normal-
ized Kahler-Ricci fl.ow converges to a Kahler-Einstein metric in the same
Kahler class as the initial metric.

THEOREM 2.51(KRF:c1::::;0 convergence). Letg(t) be a solution of the
normalized Kahler-Ricci flow, as in Theorem 2.50, with c1 (M) ::::; 0. Then
g (t) converges exponentially fast in every Ck-norm to the unique Kahler-·
Einstein metric g 00 in the Kahler class [wo].

Note that the initial-value problem for the normalized Kahler-Ricci flow
equation

a r

8tgai3 = -RaiJ + ;,gaiJ'

gaiJ (0) = g~iJ'

is equivalent to the following parabolic Monge-Ampere equation for the

metric potential function cp(x, t):

8cp det gai](x, t) r

• 8

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

t et g ai3 x, n.

(2.50)

(2.51) cp(x, 0) = 0,

where

(2.52)

and f(x, 0) is the potential function of RaiJ(x, 0) - ~gaiJ(x, 0) defined by
(2.31). To prove Theorem 2.50 and Theorem 2.51, it suffices to prove the
long-time existence of cp(x, t) and its convergence, respectively.
Presently we shall see that the reduction of the Kahler-Ricci flow to
a single parabolic Monge-Ampere equation simplifies matters considerably.
The main advantage is that it allows one to invoke the techniques of para-
bolic (and elliptic) PDE for single equations, such as maximum principles,