48 28. SPECIAL ANCIENT SOLUTIONS
of g (0) h ave a positive lower bound on /(i· By this and the assumption that
limi--+oo Ei = 0, we conclude that t here exists ki > ki-1 such that N"k' n /(i = 0 ,
which in turn implies (28.28).
Now, by again using the smooth Schonfiies theorem, for each i and j with j < i
we have
where Ki,j ~ 52 x [O, 1) has compact closure in M and where Li,j ~ 52 x [O, 1)
satisfies .Ci ,j = .Ci,j. Since M^3 ~ I!t^3 , we conclude that
(28.29)for all j < i.
Claim 2 follows from
Subclaim. n CCk; = 0.
iEN
PROOF OF THE SUBCLAIM. Fix p E B 0 k 1. By (28.29) we then have p E B 0 k , for
all i E N. Suppose t h e subclaim is false; then there exists x E C 0 k, for all i. Let "Y
b e a minimal geodesic from p to x with respect tog (0). Then "Y must pass from one
end of t he Ek;-neck. N 0 k. to t h e other end. Hence we have for i sufficiently large,1 · A ( 1 - 1 C -1
d 9 (o) (p,x) = L 9 (o) b) 2 2diam 9 (o)(JV"k,) 2 2Ek; r"k' 2 2Ek,,where c > 0 is indep endent of i. The subclaim follows easily from the fact that
-1
Ek, -t 00.Since 0 E 30 for E > 0 sufficiently small, we have for such E that any "Y E
Ray M ( 0) p asses from one end of N 0 to the other end. Recall that 'l/J 0 is the
embedding in (28.26).Claim 3. For any E > 0 sufficiently small , we have that any "Y E Ray M ( 0)
intersects 1/J 0 (S^2 x {c^1 - 4}) at exactly one point, which we define to be "Y(t-r, 0 ).
PROOF OF CLAIM 3. This follows from the facts that rays are minimizing and
that the geomet r y of any E-neck is, aft er rescaling, c-close t o that of the standard
unit cylinder (of length 2c^1 ); we leave a detailed proof of Claim 3 as a n exercise.
STEP 4. Proof of the theorem.
By the reduction to (28.25) in Step 1, it is now easy to see that (28.24) and
the theorem are conse quences of the following.
Claim 4. For any "'(1, "'(2 E R ay M (0),(28.30) lim do b 1 (t-r1 ,c) , "'!^2 (t-r^2 ,c)) = 0
c--+0 min { t-y 1 ,c, t-y 2 ,c}.
PROOF OF CLAIM 4. Since N 0 is an c-neck, we h ave that for E sufficiently smalldo b1 (t-y 1 ,c) , "'!2 ( t,, 2 ,c)) :::; 2nr" ,
where r 0 is the radius of N 0 , since t he int rinsic diameter of a round 2-sphere of
radius r is nr. Since 'Yi and "'( 2 are rays em an a ting from the same point, this
implies by the triangle inequality that
(28.31)