106 19. GEOMETRIC PROPERTIES OF K-SOLUTIONSHence, by Proposition 18.12, fixing p E M, there exist sequences {xi}~ 1
and {ri}~ 1 with ri E (0, 1] satisfying (19.39), (19.40), and in particular
R (xi, 0)--+ oo.Again let 9i(t) ~ R (xi, 0) 9 ( R (xi, 0)-^1 t). Since ~~ > 0 and 9 (t) is Kr
noncollapsed at all scales, there exists a subsequence
(Bg(O) (xi, ri) '9i (t) 'Xi)
which converges to a complete limit (M~, 900 (t), x 00 ), t:::; 0, with
(19.42) R 900 (x 00 ,0)=1 and supR 900 (·,O) ::=; 4.
Moo
Since the trace Harnack estimate is preserved under Cheeger-Gromov
limits,^24 900 ( t) satisfies the trace Harnack estimate. The limit (M 00 , 900 ( t))
must contain a line passing through x 00 (again see Theorem 18.19) and it
splits as
(Moo,9oo (t)) = (IB.xW^2 ,du^2 + 9W (t)),
where (W, 9w (t)) is a 2-dimensional oriented A;-solution (it has positive
bounded curvature by (19.42)). By Corollary 19.43, (W, 9W (t)) is a round
shrinking 2-sphere.^25
Hence, for any c > 0, there exists io E N such that for i ;::=: io, wehave that (B 9 (o) (xi,C^1 cnR-^112 (xi,O)) ,9(0)) is an embedded c-neck in
( M, 9 ( 0)) and Xi lies on the center of the embedded c-neck. Now since
d 9 (o) (xi,P) ->-oo, by choosing c = c (n) in Proposition 18.33, we have
R (xi, 0) :::; 144R (xi 0 , 0)
for all i sufficiently large. This contradicts R (xi, 0) --+ oo and the proposition
is proved. D
This proposition shows that it is not possible to have unbounded cur-
vatures at centers of c (n)-necks in noncompact manifolds with positive sec-
tional curvatures.
REMARK 19.45. There is another approach, shown to us by Yu Ding,
toward finishing the proof of Proposition 19.44 (see [54]).5. Existence of an asymptotic shrinker
In this section we derive a key estimate for the difference of the reduced
distance at two points at the same time which was not proved earlier in
this book series. After reviewing the reduced volume, we use this estimate
to address an issue which we neglected to discuss in the proof of Lemma
8.38 in Part I. This lemma was used to prove the existence of an asymptotic
shrinker.(^24) Since the convergence is in 000 on compact sets, the trace Harnack quadratics of
the sequence converge to the trace Harnack quadratic of the limit solution.
(^25) Note that since M is noncompact and has positive sectional curvature, it is ori-
entable, which implies that W is orientable.