1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. CONSTRUCTION OF GOOD COVERINGS BY BALLS 163

where c and Care defined by .A [rJ ~ ce-Cr as in (4.7), i.e.,

(4.8)

a

c ~ D · min{lo, l}n.

The strange radii is to ensure that Lemma 4.18 is true and it will later be

clear why these numbers were chosen. We easily see the following properties:

PROPOSITION 4.27 (Disjointness of smaller balls and covering of larger

balls). If k :?: K' (r) and a :S A (r), then B'k are disjoint and B (Oki r) C

Ua~A(r) Bk·

PROOF. It follows from Corollary 4.25 that

B'k C B (xk', .A [r'k]) , B (xk', 2.A [r'k]) c B'k.

The proposition then follows immediately from Proposition 4.22. D

PROPOSITION 4.28 (Bound on index of intersecting balls). For any a :?: 0

there exists an integer I (a, n) (independent of k) such that if

B~ nB'k-/= 0

for some (3 and k, then (3 :SI (a, n).

PROOF. This follows easily from Lemma 4.21. If y EB~ n B'k, then

r~ :S d(Ok,xk') + d(xk',y) + d (y,x~)

:S r'k + 5.A a + 5.A,6

(4.9) :S r'k + 10.A [OJ,

so by Proposition 4.23,

r~ :S (2a + 10) .A [OJ

since .A [OJ :S 1. By Lemma 4.21 we have (3:SA((2a+10) .A [OJ).
take I (a, n) ~A ((2a + 10) .A [OJ).

Now just

D

PROPOSITION 4.29 (Stability of the intersections of balls). There is a
subsequence {Mk, 9k} such that for every pair (a, (3) there is a number

K (a, (3) such that either B'k n Bf is empty for all k :?: K (a, (3) or B'k n Bf

is nonempty for all k :?: K (a, (3).

PROOF. Let {(ae, f3e)}eENU{O} be an ordering of the elements of the
countable set (NU {O}) x (NU {O}). We construct inductively a nesting se-

quence of subsets Ke c N. Let Ko be an infinite subset such that Br^0 n Bf^0
intersect always or never for all k E Ko. This can be done since either the
balls intersect for infinitely many k, and we take Ko to be the set of such k,
or they do not intersect for infinitely many k', and we take Ko to be the set
of these k'. Now for each subsequent£ we can construct an infinite subset Ke
of Ke-l such that B~e nBfe intersect always or never for all k E Ke. We now
consider the collection of {Ke} eENU{O} as nested subsequences of N and take
a diagonal subsequence. Define a function K : (NU {O}) x (NU {O}) -+ N