- THE NECKPINCH 51
We have l?fsl 2: 8 in the region Q2 = [x2, 1) x [t1, T). If we take 77 < 482 < 4,
then we have L(u) > 0 in Q2. Then by the maximum principle, there exists
C < oo such that la l :S Cu in Q2. Indeed, b must attain its maximum C
on the parabolic boundary of Q2. At the left end (that is at x = x2) we
have u = ?jiT/ 2: (D/64)"1, while Corollary 2.23 implies that !al is bounded
by sup la(-, O)I. At the other vertical side of Q2 (that is at x = 1) we have
a = 0. Since a is smooth, this implies a = O(s(l, t) - s(x, t)). On the other
hand, because u = ?jiT/ and Vis = - 1 at x = 1, we have limx/1 la/ul = 0 for
all t < T. Finally, at t = t1, the quantity b is continuous for x2 :S x < 1;
while we have just verified that b --t 0 as x / 1. Thus b is bounded on the
parabolic boundary of Q2, and hence bounded on Q2·
We now perform a parabolic dilation. Let B2 denote the portion of
sn+l where x 2: X2. Then we have a solution g(t) to the Ricci flow on B2
defined fort E [t 1 , T). By Lemma 2.34, this solution has Re 2: 0. Because
of spherical symmetry, B 2 is a geodesic ball in (sn+^1 ,g(t)) whose radius is
bounded from above by di ( t). Since Vi = Vi ( x2, t) 2: D / 2 on the boundary
of B2, it follows from the estimate IVisl :S 1 that the radius of B2 is bounded
from below by D /2.
Assume that the sectional curvatures of the metrics g(t) on B2 are not
bounded as t / T. Then there is a sequence of points Pk E B2 and time
tk E [t2, T) such that I Rm(Pki tk)I --t oo as k --t oo. We may choose this
sequence so that
sup IRm(Q,t)I = IRm(Pkitk)I
QEB2
holds for t :S tk· Writing Xk for the x coordinate of Pk, we note that
?f(xk, tk) --t O; and from IVisl 2: 8 we conclude that
lim dtk(Pk, P+) = 0,
k-->oo
where dt is the distance measured with the metric g(t).
Define Ek= I Rm(Pk, tk)l-^112 , and introduce rescaled metrics
1 2
9k(t) = 2 g(tk + Ekt).
Ek
Let C 1 be the constant from Corollary 2.24 for which I Rm I :S C1 ?jl-^2 holds.
Then we have
?ji(Xk, tk) :S VC1Ek;
and because l?fsl 2: 8, we also have
C1
dtk(Pk,P+) :S TEk·
The distance from Pk to the pole P + measured in the rescaled metric 9k ( 0) =
Ek,^2 g(tk) is therefore at most Cif 8. In particular, this distance is uniformly
bounded.
Translating to the rescaled metric, we find that 9k(t) is a solution of
the Ricci flow defined for t E (-Ek,^2 tk, OJ on the region B2. The Riemann
