1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. THE EXISTENCE OF ASYMPTOTIC CONES^467

Now we give a

PROOF OF THEOREM 1.26. We claim that (M, A.^2 g) converges to the

Euclidean metric cone Cone (M ( oo), d 00 ), in the pointed Gromov-Hausdorff

topology as A.-+ 0. Fix an arbitrary constant (radius) r > 0 and any con-
stant c > 0. We need to show that for A. sufficiently close to 0 the Gromov-
Hausdorff distance between the balls B>..2 9 (p, r) and Bdcone(M(oo)) (0, r) is

less than c.

We take an 160 -net (see Definition G.10)

N = {([11i] ,ai)}:, 1

of Bdcone(M(oo)) (0, r) C Cone (M (oo)), where /1i E Ray M (p) and ai E (0, r].

Then

( )

c c

d GH B dcone(M(oo)) ( ' ) ' Or N <-<-- 10 - 3 ·

Let

N>.. ~ { /1i (A.-^1 ai)} : 1 c B>..2 9 (0, r) = B 9 ( 0, A.-^1 r) c M.

We claim that it follows from (1.8) that

(I.11) dGH ( N>.., LJiERayM(P)/1 ([o, A.-

1

r])) :S ~

for A. sufficiently small, where UiERayM(p)f1([0,>..-^1 r]) c B>..2 9 (p,r). To

prove (I.11), it suffices to show that for any

/1 (>..-^1 a) E LJiERayM(p)/1 ([o, >..-^1 r])

there exists /1i (A. -l ai) E N >.. such that

(I.12)

Note (b], a) E Cone (M ( oo )). Since N is an { 0 -net of Bdcone(M(oo)) (0, r),

there exists ([!1i] , ai) E N such that

c

dcone(M(oo)) ((bi], ai), ([11], a)) :S 10 ·

This implies (I.12) for A. sufficiently small in view of (I.8).

Now from (I.8) again, we have

c

dGH (N, N>..) :S 3

when A. is sufficiently small.

Recall that given any co> 0, there exists T > 0 such that fort> T and

x in the sphere S(p, t) there exists f1 E Ray M (p) such that

d(x,11(t)) ------'---'--''---'--:...:... < co -

t - 2

(see Lemma 7(i) in [43] for example). This implies that

dGH ((UiERayM(p)f1([o,A.-^1 r])) ,B>..2 9 (0,r)) :S ~