1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

160 4. PROOF OF THE COMPACTNESS THEOREM

3.2. Choice of ball centers. In this subsection we will find centers

for the balls which will make up the necessary covers. Define >. [r] as

(4.7) /\ ' [ ]. 1 [ r =;= Dμ r, lo ] = D a ·mm. { lo, 1 }n · e -Cr ,

where D = D (n, lo) is a large constant, to be chosen later, depending only

on n and the uniform lower bound lo for inj (Ok). Note that >. is a decreasing

function of r. We shall choose D large enough so that >.[OJ ::::; 1.

We shall work on an individual manifold (Mn, g, 0) and then apply the

result to (M'k, 9k, Ok). Choose a sequence of points {x°'}~=O in M with

N E NU {O, oo}, which we call a net, as follows. Let x^0 = 0 and r^0 ~

d (x^0 , 0) = 0. Let

51 ~ {x EM: B(x,>.[d(x,O)]) nB (x^0 ,>. [r^0 ]) = 0}.

If 51 is empty, then B (x^0 ,2>. [r^0 ]) = M and we take N = 0 and stop

choosing points. Since the balls in the definition of 51 are open, 51 is a

closed set. Hence, if 51 is nonempty, then there exists a point x^1 E 51 such

that r^1 ~ d ( x^1 , 0) = d ( 51 , 0). Since x^1 E 51 , we have

B ( x^1 ' A [r^1 J) n B ( x^0 ' A [r^0 ]) = 0.

Next let

52 ~ { x EM : B (x, >. [d (x, O)]) n B ( x,6, >. [r,6]) = 0 for f3=0,1}.

Again, since 52 c 51 is a closed set, if 52 is nonempty (if 52 is empty, we

take N = 1 and stop), we may choose x^2 E 52 such that r^2 ~ d ( x^2 , O) =

d (5^2 , 0). We have

B ( x,6, >. [r,6]) n B (x'Y, >. [r1]) = 0 for f3=/='YE{O,1, 2}.

By induction, assuming that the points x^0 , x^1 , x^2 , ... , xa-l have been cho-

sen, let

5°' ~ { x E M : B ( x, >. [ d ( x, 0)]) n B ( x,6, >. [ r,6]) = 0, f3 = 0, ... , a - 1}.

Note that S°' c 5a-^1. If 5°' is nonempty (if 5°' is empty, we take N = a-1
and stop), choose x°' E 5°' such that r°' ~ d(x°',O) = d(5°',0). Then

B ( x,6, >. [r,6]) n B (x1, >. [r^7 ]) = 0 for f3=/='YE{O,1, 2, ... , a}.

LEMMA 4.21 (Existence of covers with bounds on the number of balls).

Let (Mn, g) be a complete Riemannian manifold with sectional curvatures

IKI :S; Co and injectivity radius inj (0) 2': lo> 0. For each r > 0 there exists

a nondecreasing function A (r) such that the finite collection

{B (x°', 2.\ [r°']) : 0::::; a::::; A (r)}

forms a cover for B (0, r) and r°' > r if a > A (r). Furthermore, we can

choose A (r) to depend only on n, r, Co and lo (in particular, not depend on

the manifold M).