1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

168 4. PROOF OF THE COMPACTNESS THEOREM

Hence we have the following local property which is a key step to Proposition

4.33.

PROPOSITION 4.34 (Local maps converge to the identity in a sense). If

a and f3 are such that Bk n B~ I- 0 for k sufficiently large, then the maps

Ffd,/3 : Ef3 -----+ EJ!3 converge to the identity (inclusion) map id13 as k, £-----+ oo.

Now we proceed to average the local maps to construct a map on a large

ball. To apply Proposition 4.53 on averaging maps, we need to construct

a partition of unity subordinate to the covering {Bk} a:SA(r) of B (Ok, r) as

follows. For a:::; A (r) let 'l/J°' be a smooth function which is 1 on E°' CE°'

and 0 outside of E°' (E°' = (Hf:)-^1 Bk). We construct a partition of unity

on B (Ok, r) by letting

{

· 'ljJ°'o(Hk)-^1 (x). a

a. I _ 1 If X E Bk,

'Pk (x) =;:: ~1'.SA(r) 'I/JI o (Hk) (x)

O if x ~Bk,

where a :::; A (r). By Lemma 4.18(4) the denominator is no less than 1

and by Lemma 4.18(5) the number of terms in the denominator which are

well defined is bounded by I (n, Co), independent of k. To ensure that the

basepoint is preserved, we need the partition of unity function ( <P1k below)

indexed by a I- 0 to vanish in a neighborhood of Ok, so we introduce a C^00

function x : E^0 -----+ IR such that x = 0 in a neighborhood of the origin and

x = 1 outside (Hg)-^1 .Bg (note that this set is a Euclidean ball independent

of k). Define Xk : Mk -----+ IR by Xk (x) 4:: x o (Hg)-^1 (x) if x E B2 and

Xk ( x) 4:: 1 otherwise. Then we take

</>1k : Mk -----+ IR

if x E Bk,

if x ~Bk,

while for a = 0,

The collection of functions { ¢1k} a:SA(r) is a partition of unity subordinate

to the covering {Bk} a:SA(r) of B (Ok, r).