- THE NECKPINCH 47
( 2) There exists a time T bounded above by r min ( 0)^2 / ( n - 1) such that
the radius rmin(t) of the smallest neck satisfies
(n - l)(T - t) :S rmin(t)^2 :S 2(n - l)(T - t).
(3) The solution is concave ( 7./Jss < 0) on the polar cap.
(4) If the right-most bump persists, then D ~ limvriP(x*(t), t) exists.
If D > 0, then no singularity occurs on the polar cap.
We begin with the observation that the number of necks cannot in-
crease with time, and further that all bumps/necks will be nondegenerate
maxima/minima unless one or more necks and bumps come together and
annihilate each other.
LEMMA 2.31. Let g (t) : 0 :S t < T be a solution of the Ricci flow of
the form (2.43). At any time t E (0, T), the derivative v = 7./Js has a finite
number of zeroes when regarded as a Junction of x E ( -1, 1). This number
of zeroes is nonincreasing in time. Moreover, if 'ljJ ever has a degenerate
critical point, the number of zeroes of 1/Js drops.
PROOF. The derivative v = 7./Js satisfies (2.54), which can be written as
a liner parabolic equation Vt = Vss + Qv, with
w l - v^2
Q = (n - 2) 'ljJ + (n - 1) - 1/J-.
Since gs = ~ ffx, one can in turn write (2.54) as
Vt= <p-^1 (<p-^1 vx)x + Q (x, t) v =A (x, t) Vxx + B (x, t) xv+ C (x, t) v
for suitable functions A, B, C. Since v ---+ =fl as x ---+ ±1, the Sturmian
theorem [5] applies. 0
We next derive upper and lower bounds for the rate at which a neck
shrinks. These estimates show that a singularity will develop at the smallest
neck in finite time, unless the solution loses all its necks first.
LEMMA 2.32. Let g (t) be a solution to the Ricci flow of the form (2.43)
such that 1 7./Js I :S 1 and R > 0 initially. Then
( n - 1) ( T - t) :S r min ( t)^2 :S 2 ( n - 1) ( T - t).
In particular, the solution must either lose all its necks or else become sin-
gular at or before
T = rmin (0)
2
n - l
PROOF. Note that 'ljJ (·, t) is a Morse function except perhaps for finitely
many times, and that rmin (t) is a Lipschitz continuous function. We claim
that
n-l d n-l
- < -r · (t) < ---
rmin (t) - dt mm - 2r min(t)
holds for almost all times. The lemma follows from the claim by integration.