1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. PROOF OF THE STABILITY OF HYPERBOLIC LIMITS 229

To see (33.26), first note that by (33.24) we have

(33.28)

for all i. Moreover, for i large enough

(33.29)

by (33.22) and since qii : (fh h) ---+ (Vi, g (f3i)lv) is as close to an isometry as we

like in any Ck-norm.^10 Thus qi;^1 o Fi (f3i), as measured from h to h, is close to an

isometry in its domain.

Since HA is compact, it follows from (33.28) and (33.29) for our given k, that

qi;^1 o Fi (f3i) (HA) c B (:i\'°, C)

for some C < oo independent of i. Since the sequence {U°diEN exhausts il, we have

that for i large enough, qi;^1 o Fi (f3i) (HA) C Ui C il; that is,

Fi (,Bi) (HA) c Vi c M,

which is (33.26). Since Fi (f3i)-^1 (Vi) =HA, we have that (33.29) says

(33.30) II (qi;

1

o Fi(f3i))* h - hl1iA llck(HA,h) :S ~-

Now consider the embeddings

<I.>:--^1 oF; (/3;) - -

H A ' ui c H

F;(/3; ) \. ,/<I!;

M.

By (33.30) and the "Arzela- Ascoli theorem for maps" (see Claim 2 in the proof of

Proposition 33.14), we have that { qi;^1 o Fi (f3i)} subconverges in ck-^1 (HA, h) to

a map W 00 : HA---+ il.

Moreover, by (33.22) we have

(33.31)

Now qi; g (f3i) converges to h in C^00 on compact sets, while we only know that

qi;^1 o Fi (,Bi) converges to W 00 in Ck-^1 (HA, h). However, we claim that by passing

to a subsequence, the latter convergence is also in C^00 (HA)· Because of this, we

(^10) Note that (Fi (,Bi)-^1 r his close tog (,Bi)·