- INTRODUCTION TO THE KAHLER-RICCI FLOW 75
r.p ( t) defined on all of M such that(2.27)By (2.6) we have_ _ _ _ det ( g~ 8 + ta'Ya" log det gz 0 + a,,a"r.p ( t))
Raj3 (t) - Ra/3 (0) - -8a813 log 0 ·
det g'YJHence, by differentiating (2.27), we obtain8a8j3 ( :t r.p) = -Ra'iJ - 8aB13 log det g~J
_ _ det ( g~J + t8'Y8o log det gR 0 + 8'Y8ar.p ( t))
-8a813log d 0.
et g'YJHence we conclude that the Kahler-Ricci flow equation on a closed mani-
fold is equivalent to the following parabolic (scalar) complex Monge-
Ampere equation:
8r.p det ( g~J + t8'Y8o log det gZ 0 + 8'Y8 0 r.p ( t))
at =log det go- + c1 (t)
'Yo(2.28)for some function of time ci (t). By standard parabolic theory, given any C^00
initial function r.p 0 on a complete Kahler manifold with bounded bisectional
curvature, there exists a unique solution r.p (t) to (2.28) with r.p (0) = r.p 0 ,
defined on some positive time interval 0 :s; t :s; c:. We also have the following.
LEMMA 2.36 (The Kahler property is preserved under the Ricci flow).If (Mn, J, go) is a closed Kahler manifold, then there exists a solution to
the Kahler-Ricci flow g (t), 0 :s; t :s; c:, for some c: > 0 with g (0) = go.
Furthermore g (2t) is a solution of the (Riemannian} Ricci flow. Also any
solution g (t) of the (Riemannian) Ricci flow with g (t) =go must be Kahler
(preserving the compatibility with the almost complex structure).PROOF. Given go, we can find a solution r.p (t), 0 :s; t :s; c:, of (2.28) with
c 1 (t) = O. From the derivation of (2.28), we know that g (t) defined by (2.27)
is a solution of (2.26). Hence g (2t) is a solution of the Ricci flow. The last
statement follows from the uniqueness of the initial-value problem for the
Ricci flow. D
REMARK 2.37. From the derivation of (2.28) it is clear that if we have
a bounded C^4 -solution r.p (t) for some c1 (t) on any complex manifold (re-
gardless of completeness and compactness), then we get a C^2 -solution g (t)
defined by (2.27) to the Kahler-Ricci flow.