1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

4. NECKS IN MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE 65

Let A be a constant to be chosen later, and E ( n) will be chosen so that

it satisfies c(n)-^1 - 1 >A. ·

First we prove that the Busemann function satisfies

b'Y (q) < bo for any q E Nlnn - 'I/; (sn-l x (-A, 0))

provided A 2:: i7r. For s sufficiently large, any unit speed minimal geo-

desic /3s (t) joining /3s (0) = q and /3s (ts)= 1(s), where ts~ d(q,1(s)),

must intersect 'I/; (sn-l x {u}) for each u E [-A,c(n)-^1 -1]. In particular

there exist to E [O, ts] such that /3s (to) E 'I/; ( sn-l x {O}). By the triangle

inequality we have

s - d ( / ( s) , q) = s - d ( / ( s) , /3 s (to)) - d (/3 s (to) , q)

:S s - d (! (s), qo) + d (/3s (to), qo) - d (/3s (to), q)

(18.52) :S b'Y (qo) + d (/3s (to), qo) - d (/3s (to), q).

Recall that we have earlier set en ~ J(n -1) (n - 2). If c(n) is small

enough, since the geometry of the neck S)1 is close to that of the standard

cylinder, we have R (qo) > 0 and^16

(18.53)

and

(18.54)

Hence for A 2:: i1f, by (18.52), (18.53), and (18.54), we have

s - d (r (s), q) :S b'Y (qo) - :

5

1fcnR (qo)-^112.

Thus for q E Nlnn -'I/; (sn-l x (-A,O)), where A 2:: i1f,

(18.55) b'Y (q) = lim (s - d (r (s), q)) :S b'Y (qo) -

1

2

5

1fcnR (qo)-^1 /^2 < bo.

s-+oo

Second we prove that

b'Y (q) > bo for any q E Nout - 'I/; (sn-l x (0, A))

provided A 2:: 51f. For s sufficiently large, any unit speed minimal geodesic

T/s (t) joining T/s (0) = qo and T/s (ts)= 1(s) intersects 'I/; (sn-l x {A}) at

T/s (ti). Let ( (t) be a unit speed minimal geodesic joining ( (0) = qo and

( ( tq) = q. The minimal geodesic ( intersects 'I/; ( sn-l x {A}) at ( ( t2).

Consider the triangle b..'f/s (t1) qo( (t2); since the geometry of the neck SJt is

(^16) The points f3s (to) and q 0 lie on the same center sphere. For the standard (round)
cylinder sn-l (r) x JR of radius r we have R = c;,r-^2 ; note that diam (sn-l (r)) = 7rr =
1fCnR-1/2.