1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

172 4. PROOF OF THE COMPACTNESS THEOREM

Next we turn to prove that Fk£;r is a diffeomorphism. Since F[k is the

inverse of Ffj, by Proposition 4.34, for each f3 we have Fk£,/3 --+ id13 and

Fek,f3 --+ id13 when k, f, --+ oo. Then by a simple argument using Proposition

4.54 we conclude that Fek roFke rand Fke roFek r both approach the identity

map when k, f, --+ oo. It follows ' ' that Fke,r ' is invertible. '

Now it follows from the inverse function theorem and Proposition 4.35

that Fi:/r is an (c',p)-pre-approximate isometry. Hence, given (c,p) and r,

there is~ ko such that Fk£;r is an (c,p)-approximate isometry fork,£ 2: ko.

Proposition 4.33 is proved.

4.3. The directed system. We are now in a position to construct a

directed system whose direct limit will give us the limit manifold (M~, g 00 ).

We first show that, after passing to a subsequence, the existence of approx-

imate isometries whose compositions are also approximate isometries, as

close as we like to isometries.

PROPOSITION 4.39 (Metrics are almost isometric on large balls). There

exists a subsequence { (M1k., 1 9k.) 1 } such that for any c > 0 and p E N

jEN

there exists Jo= Jo (c,p) EN such that if J >Jo, then there exist maps

Wj : B ( okj' 2j) --+ B ( okj+l? 2H^1 )

with

wj ( okj) = okj+i

such that for any f, E N the composition map

Wj,£ ~ WjH-1 O•. ·OWj+l oWj : (B ( okj? 2j) '9kj) --+ ( B ( okHf' 2H£) '9kj+f)

is an (into) (c,p)-approximate isometry.

PROOF. We may assume that Cj is increasing as J increases and that

Co 2: 1. We shall inductively define the subsequence {(Mk., 1 9k 1 .) }. It

jEN

is sufficient to construct a sequence {wj}jEN such that Wj is a ( c;^1 2-j,J)-

approximate isometry. In this case we can use Corollary 4.8 to see that

Wr,£ is a (Cr 2=~!;-^1 c;-^1 2-i, r )-approximate isometry. In fact, since Cj is

increasing in J, we have

r+£-1 oo oo

Cr L c;-12-i ::::; Cr L ci-12-i ::::; L 2-i = 21-r'

i=r i=r i=r

which implies that Wr,£ is a (2^1 -r, r )-approximate isometry. We also have by

Proposition 4.4 that '1Fo(B(Ok 0 ,l)) c B (ok 1 , (1+Cc)^1 )

112
) c B(Ok 1 ,2)
since