1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

140 3. THE COMPACTNESS THEOREM FOR RICCI FLOW

(2) (gk o Jk) (X, Y) ~ 9k (JkX, JkY) = 9k (X, Y) for all X, YET Mk,

(3) 'hJk = 0.

Since {(Mk, gk, Ok)} converges to (M 00 , g 00 , 000 ), there are diffeomor-

phisms <Pk : Uk ---+ Vk such that <P'kgk ---+ g 00 uniformly on compact sets. From
(1) we know (k^1 L o Jk o (<Pk), as (1, 1)-tensors, are uniformly bounded
on any compact set K C M 00 with respect to the metrics <P'k9k· Since the
<.P'kgk are equivalent to g 00 on K, we conclude that ( <.Pk^1 L o Jk o (<Pk), as
(1, 1)-tensors, are uniformly bounded on any compact set K C Moo with
respect to the metric 900.
Note that (3) is equivalent to \i'if>h,gk ( ( <.Pk^1 L o Jk o (<Pk)*) = O; hence

( \i'if>h,gk y ( ( k^1 L o Jk o ( k)*) = o for all p.

From the proof of Corollary 3.15, there exists a subsequence ( <.Pk^1 ) o Jk o
(<Pk) converging in C^00 , as a (1, 1)-tensor on compact sets, to a smooth
map J 00 : T M 00 ---+ T M 00. Since

(1') 0 = ( ( cpkl L 0 Jk 0 (<Pk)*)

2

+ idTM 00 ---+ J! + idTM 00 '

(2') o = ( <t>'kgk) o ( ( <t>k^1 L o Jk o ( <t>k)*) - <t>'kgk ---+ 900 o Joo - 900,

(3') 0 = \i'if>i,gk ( ( <.Pk^1 L O Jk O (<Pk)*) ---+ \7 ooJoo,

we have J! = -idTM 00 , g 00 o J 00 = g 00 and \7 00 J 00 = 0. We conclude.that
(Moo, 900, J 00 ) is a Kahler manifold. D
Applying Theorem 3.10, we obtain the following corresponding result
for the Kahler-Ricci fl.ow.

THEOREM 3.23 (Compactness theorem for the Kahler-Ricci flow). Let

{ (M~n,gk (t), Ok)}, t E (a,w) 3 0, be a sequence of complete pointed solu-

tions to the Kahler-Ricci flow of complex dimension n. Suppose

IRmklk::; Co on Mk x (a, w)

for some constant Co < oo independent of k and that

inj 9 k(o) (Ok) 2': lo

for some constant lo > 0. Then there exists a subsequence of solutions

such that { (Mk, 9k ( t) , Ok)} converges to a complete pointed complex n-

dimensional solution to the Kahler-Ricci flow (M~, g 00 (t), 000 ) , t E (a, w),

as k---+ oo.

PROOF. By Theorem 3.10, there exists a subsequence which converges

to a Riemannian solution (M 00 , g 00 (t), 000 ), t E (a, w), to the Ricci fl.ow.

From the previous theorem, (Mk, 9k (0), Jk) converges to (M 00 , 900 (0), Joo)

as Kahler manifolds for some complex structure J 00 • Now by assumption,
9k (t) remains Kahler with respect to Jk. Hence 9k (t) o Jk = 9k (t) and
\7 gk(t)Jk = 0 for all t E (a, w), which implies g 00 (t) o J 00 = g 00 (t) and

\7 900 (t)Joo = 0 for all t E (a,w). That is, g 00 (t) remains Kahler with respect

to J 00 • D