1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

156 4. PROOF OF THE COMPACTNESS THEOREM

converges in C^00 uniformly on compact sets to a C^00 diffeomorphism <:1? 00 :

u---+ v.

PROOF (SKETCH). Let {xa} and {ya} be the standard Euclidean coor-

dinates on U and V, respectively. Since <Pk are isometries, we have

( 4.4)

a (<Pkt a (<Pk)/)

(gk)ab = axa axb (hk)a/).

Thus

_ a (<Pkt a (<Pk)/) (h ) ( )ab

n - axa axb k a/) 9k

and the partial derivatives^8 ~;J"' are all bounded. Thus there is a sub-

sequence of {<Pk} which converges to a map <:1? 00 • By symmetry the same

argument applies to { <J?/;^1 }; hence <]? 00 is invertible.

Taking the derivatives of both sides of ( 4.4), we get that

a a^2 (<Pkt a (<Pk)/) a (<Pkt a^2 (<Pk)/)

axe (gk)ab = axeaxa axb (hk)a/) + axa axeaxb (hk)a/)

+ a (<Pkt a ( <Pkl a (<Pk)'_!!_ (hk).

axa axb axe ay'Y a/)

From this equation we can express 8;;~81: as a polynomial function of

( 9k ) ab ' ( gk -l)ab ' (h k ) a/) ' (h-1)a/3 k ' axe^8 ( 9k ) ab ' oy'Y^8 (h k ) a/) an d axa-8(<Pk)"' usmg ·

symmetry in the usual way (see §5 of [187] for the explicit formula). Thus

I

82(<J> axctfxa )"'I can be bounded. By differentiating the formula for f32(<J> axcrfxa )"' and

using induction, we can bound all higher derivatives of (<Pk)a. This implies

the corollary. D

2.3. Review of direct limits. Let {(Ak, fk)}kEN be a sequence of

topological spaces and open embeddings:

A 1 ---+ Ii A 2 ---+ h... ---+ A k ---+ !k A k+ 1 ---+ •...

Consider the compositions

fk.e_ ~ f.e_-1 ° f.e_-2 ° · · ·^0 fk+1 ° ik : Ak ---+ A.e_

defined fork :S .£,where fkk ~ idAk : Ak---+ Ak, the identity map. Clearly

hm o fk£ = fkm

for all k :S .e :S m. That is, ( { Ak}kEN, {fk.eh:::;.e) is a directed system of

topological spaces (see Definition 15.1 in [159]). We will use II to denote
disjoint union.

DEFINITION 4.12 (Direct limit). The direct limit is

limAk = (IIkAk)/ rv,

----+