1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. THE LIMIT MANIFOLD (M~,g 00 ) 171

where I· I is the Euclidean norm and '\7 is the Euclidean covariant derivative

{i.e., partial derivative).

PROOF. Let x = {xi} be the standard Euclidean coordinates on U and

let cPk (x) = (¢% (x)):=l. Since cPk ----t id in C^00 uniformly compact sets, we

have that ~!~ ----t of and ?a:~~ ----t 0 uniformly on K where a is multi-index

with lal 2: 2. Since hk ----t h 00 in C^00 uniformly compact sets, we have that

(^8) °'(hk(x))ab ----t (^8) °'(hoo(x))ab uniformly on K for any a Now
(Bx)°' (Bx)°' ·
cPkhk * - hoo = ( (hk)ab 8¢k, {)xi 8¢~ {)xj - (hoo)ij ) ----t 0
uniformly on K. So the lemma holds for p = 0.

For any r > 0,

'\7r (¢khk - hoo) = c:x~a ((hk)ab ~!~ ~~~ -(hoo)ij)) lalsr

(

{)a (hk)ab 8</Jk, 8¢~ 3a (hoo)ij )

= (axt. {)xi axj - (axft + ea,i,j '

where 8a,i,j is a sum of terms of the form

3a1 (hk)ab 3a2 8</Jk, 3as 8¢~

(ax ti (ax t^2 {)xi (ax t^3 {)xj

with la1l+la2l+la3I =rand la2I 2: 1. Hence '\7r (¢khk - g 00 ) ----t 0 uniformly
on K. The lemma is proved. D

With this we are ready to prove the following.

LEMMA 4.38 (Fke,r is an (<s,p)-pre-approximate isometry). For any E > 0

and p > 0 there exists "" = "" ( E, p) such that

l'\7ik (Fk'e;rge - gk) l 9 k ~ E

for all q ~ p if k, £ 2: ""(c,p). Hence Fke,r is an (c,p)-pre-approximate

isometry.

PROOF. We work in a coordinate chart ( Ef3, Hf). By Proposition 4.35,

for any E > 0 there exists ko = ko ( r, E) such that I '\7q ( G%e;r - idf3) I < E if

k,£ 2: ko.

By Proposition 4.32, the metrics g~ = (Hf)* gk are uniformly equiv-

alent in the C^00 - norm to the Euclidean metric on Ef3. Thus it suffices to
estimate the partial derivatives of Fk'e·rge - gk using the Euclidean metric.

Since G%e-r ----t idf3 and gf ----t g~, we ~ay use Lemma 4.37 to conclude that

( G%e;r) *;ff ----t g~ in the C^00 -Euclidean norm as k, £----too. The desired es-

timates now follow from the fact that g% ----t g~ in the C^00 -Euclidean norm
as k ----t oo. D