52 2. SPECIAL AND LIMIT SOLUTIONS
curvature of 9k is uniformly bounded by I Rm I :S 1, with equality attained
at Pk at t = 0. One may then extract a convergent subsequence whose
limit is an ancient solution 900 (t) of the Ricci flow on JRn+l x (-oo, OJ with
uniformly bounded sectional curvatures. The limit solution has nonzero
sectional curvature at t = 0 at some point P* whose distance to the origin
is at most Cif 8.
We introduce the radial coordinate r = r(x, tk) = Ek"^1 (s(l, tk)- s(x, tk)).
Then the metric 9k(O) = Ek"^2 9(tk) seen through the exponential map at the
pole P + is given by
9k(0) = (dr)^2 + Wk(r)^2 g, with Wk(r) = Ek^1 'lf;(s(l, tk) - Ekr, tk)·
The metrics 9k converge in C^00 on regions r :S R for any finite R, and the
functions wk hence also converge in C^00 on any interval [O, R]. The scale
invariant quantity a is given by a = 'l/J'l/Jss - 'l/J; + 1 = WWrr - w; + 1, and
it satisfies lal ::::; C'lf;T/ ::::; CEZwz. Thus we find that the limit Woo = lim wk
satisfies
WWrr - w; + 1 = 0.
Hence for some >., μ, one hasw(r;>.,μ) = {*sin>.(r - μ),
r-μ>. < 00>. = 00
Since -Wr = -'l/1 8 E [8, 1] cannot vanish, the only valid solution is the one
with >. = oo and μ = 0, that is w 00 (r) = r. But then the limiting metric
900 = lim9k(O) is (dr)^2 +r^2 g; to wit, 900 is the fiat Euclidean metric. Because
this is impossible, we conclude that the sectional curvatures of the metrics
9(t) are in fact bounded from above for all x E (x2 (t), 1) and t < T. Since
Corollary 2. 24 bounds the sectional curvatures for x E (x* (t), x2 (t)), we
have proved Lemma 2.35. D
5.4. Convergence to a shrinking cylinder. In the third part of our
analysis, we derive estimates which indicate that a neckpinch asymptotically
approaches the shrinking cylinder soliton (2.41). By the construction in
Subsection 5.5 below, we may assume that the solution keeps at least one
neck. Then letting x _ (t) and x+ (t) denote tile left-most and right-most
bumps, respectively, we define the 'waist'W (t) = [x_ (t), X+ (t)J
for all times t such that the solution exists. The key to this part of our
analysis is the quantity. Ko K
F =;= - Ki logK1 = L logL.
Notice that Fis positive in a neighborhood of a neck. An application of the
maximum principle will let us bound F from above when it is positive and
K1 = L is sufficiently large. The value of this estimate is that the factor