74 2. KAHLER-RICCI FLOWSturm [304]. The existence of Kahler-Einstein metrics on complete non-compact manifolds with c 1 < 0 and c 1 = 0 has been well studied (see Cheng
and Yau [94] and Tian and Yau [349], [350] for example). For work on
the existence of singular Kahler-Einstein metrics on certain classes of closed
Kahler manifolds where c 1 does not have a sign, see Tsuji [360] (for some
further recent work see Cascini and La Nave [60] and Song and Tian [337]).
Kahler-Einstein metrics are closely related to Kahler-Ricci solitons and
hence the Kahler-Ricci fl.ow which will be discussed next in this chapter.
4. Introduction to the Kahler-Ricci flow
In this section we introduce the Kahler-Ricci fl.ow system and its equiva-
lent formulation as a single parabolic Monge-Ampere equation. We discuss
some basic estimates which may be proved using the maximum principle.4.1. The Kahler-Ricci fl.ow equation. Let (Mn, J) be a closed
manifold with a fixed almost complex structure. Given a Riemannian metric
g, we may define a 2-tensor w by w (X, Y) ~ g (JX, Y). Recall that when g
is Hermitian, w is antisymmetric (i.e., defines a 2-form) and w is called the
Kahler form. If a solution g (t) to the Ricci fl.ow gt9ij = -2~j is Hermitianat some time t, then w satisfies the equation gtw = -2p at that time, where
p = p (t) is the Ricci form of g (t). Hence
:t ( dw) = d ( :t w) = -2dp = 0
whenever g (t) is Kahler (we can define even when g is not Hermitian). This
suggests that if g (0) is Kahler, then under the Ricci fl.ow g (t) is Kahler for
all t 2 0.
Consider the Kahler-Ricci fl.ow equation
f)
(2.26) f)t9ai] = -Rai3
for a 1-parameter family of Kahler metrics with respect to J, which is ob-
tained from the Ricci fl.ow by dropping the factor of 2. Now we derive the
parabolic complex Monge-Ampere equation to which the Kahler-Ricci fl.ow
is equivalent. For a complete initial Kahler metric with bounded curvature,
we will use this scalar equation to prove the short-time existence of a so-
lution to the initial-value problem for the Kahler-Ricci fl.ow. On a closed
manifold we will use the scalar equation to prove that the Kahler property of
an initial metric is preserved under the Ricci fl.ow and to prove the long-time
existence of solutions to the Kahler-Ricci fl.ow.
By (2.26), gt [w] = - [p (t)] = - [p (O)], so that the Kahler class of the
metric at time t evolves linearly,
[w (t)] = [w (O)] - t [p (O)],
and the real (1, 1)-forms w (t) - w (0) + tp (0) are exact fort 2 0. Let g~i3 ~
9ai3 (0). Using Lemma 2.26, for each t there exists a real-valued function