138 3. THE COMPACTNESS THEOREM FOR RICCI FLOWWe shall show that there are metrics g 00 ( t) , for t E (a, w) , such that
900 (0) = 900 and {(Mb 9k (t), Ok)} converges to (Moo, 900 (t), Ooo) in C^00 •
Since {(Mk, 9k (0), Ok)} converges to (Moo, 900, Ooo), there are maps
<I>k : Uk -+ Vk such that <I>'kgk (0) -+ g 00 uniformly on compact sets. We shallapply Lemma 3.11 with to= 0, g 00 as the background metric, and 'kgk (t)
as the sequence of metrics. Note that the assumptions of the lemma are
satisfied at t 0 = 0 because convergence assures us that the 'kgk (0) converge
to g 00 uniformly on compact subsets. We can now apply Corollary 3.15 with
g = g 00 + dt^2 on M 00 x (a, w) to find a subsequence of 'kgk ( t) + dt^2 which
converges to g 00 (t) + dt^2 in C^00 on compact subsets. (Since gt is orthogonal
to vectors on M 00 in each metric 'kgk ( t) + dt^2 , it is orthogonal to them
in the limit metric.) Hence there is a subsequence {(Mk, 9k (t), Ok)} which
converges to (M 00 , g 00 (t), 000 ), where g 00 (t) is defined to be the limit of
<l>'kgk ( t). Since all derivatives of the metric converge, the Ricci curvature
of 9k (t) converges to the Ricci curvature of g 00 (t) and hence the limit is a
solution of the Ricci fl.ow. This concludes the proof of Theorem 3.10.3. Extensions of Hamilton's compactness theorem
In this section we give several variations of Theorem 3.10.3.1. Local compactness theorems. From the proof of the compact-
ness Theorem 3.9 given in the next chapter and the proof of Theorem 3.10
given above, without too much difficulty one sees that a local version of
Theorem 3.10 holds. In particular, we have the following.
THEOREM 3.16 (Compactness, local version). Let {(Mk, 9k (t), Ok)}kEN,
t E (a, w) 3 0, be a sequence of complete pointed solutions to the Ricci flow.If there exist p > 0, Co < oo, and lo > 0 independent of k such that
IRmklk:::::; Co in Bgk(o) (Ok,p) x (a,w)
and
injgk(o) (Ok) ~lo,
then there exists a subsequence such that { ( B gk(O) (Oki p) , gk( t) , Ok)} kENconverges as k -+ oo to a pointed solution (B~, g 00 (t), 000 ), t E (a, w),
in C^00 on any compact subset of B 00 x (a, w). Furthermore B 00 is an open
manifold which is complete on the closed ball Bg 00 (o) (0 00 ,r) for all r < p.
EXERCISE 3.17. Prove Theorem 3.16.
A simple consequence of Theorem 3.16 is (see [186]) the following corol-
lary.
COROLLARY 3.18 (Compactness theorem yielding complete limits). Let
{(Mk,gk(t),Ok)}kEN' t E (a,w) 3 0, be a sequence of complete pointedsolutions to the Ricci flow. Suppose for. any r > 0 and E > 0 there exist
constants Co (r, E) < oo such that
!Rm klk:::::; Co (r,r::) on Bgk(o) (Ok, r) x (a+ r::, w - r::)