1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. EXTENSIONS OF HAMILTON'S COMPACTNESS THEOREM 139

for all k EN. We assume inj 9 k(o) (Ok) 2-: lo for some lo > 0. Then there exists

a subsequence {(Mk, gk (t), Ok)} which converges to a complete solution to

the Ricci flow (M~, g 00 (t), 000 ), t E (a, w).

REMARK 3.19. Note that the limit solution (M~, g 00 (t), 000 ) may not

have bounded curvature.

Without the injectivity radius estimate, we may use the trick of locally

pulling back the solutions by their exponential maps (since the pulled-back

solutions satisfy an injectivity radius estimate). We have the following.

COROLLARY 3.20 (Local compactness without injectivity radius esti-

mate). Let {(Mk', gk (t), Ok)}kEN, t E (a, w) 3 0, be a sequence of complete

solutions to the Ricci flow with

IRmklk:::::; Co in B 9 k(o) (Ok,P) x (a,w).

Then there exists a subsequence such that

{ ( BTokMk (o, c) ' ( exp~k(O))* gk(t)) 'oLEN' where c ~min {p, 7r/vCo}'

converges to a pointed solution (J3~,g 00 (t),0 00 ), t E (a,w), on an open

manifold which is complete on the closed ball B 900 (o) (0 00 , r) for all r < c.

REMARK 3.21. There is a similar result for geodesic tubes; see §25 of
Hamilton [186].

3.2. Compactness for Kahler metrics and solutions. Without
much difficulty, the compactness theorems apply to Kahler manifolds and
solutions of the Kahler-Ricci flow (see also Cao [48] and Theorem 4.1 on
pp. 16-17 of Ruan [314]).

THEOREM 3.22 (Compactness for Kahler metrics). Let { (M~n, gki Ok)}

be a sequence of complete pointed Kahler manifolds of complex dimension n.

Suppose

IV'~ Rmklk:::::; Gp on Mk

for all p 2-: 0 and k, where Gp < oo is some sequence of constants independent

of k, and

inj 9 k (Ok) 2-: lo

for some constant lo > 0. Then there exists a subsequence {jk}kEN such that

{(Mjk,gjk,Ojk)}kEN converges to a complete pointed complexn-dimensional
Kahler manifold (M~, g 00 , 000 ) as k ---* oo. See the ensuing proof for the
meaning of the convergence of the complex structures Jk --t Joo.
PROOF. Since Kahler manifolds are Riemannian manifolds, we can ap-
ply the Cheeger-Gromov Compactness Theorem 3.9 to obtain a pointed
limit (M~, g 00 , 000 ) which is a complete Riemannian manifold. So the
only issue is to show that the limit is Kahler. Let Jk denote the complex
structure of (Mk, gk). We have for each k E N,

(1) Jf = -id™k'