1547671870-The_Ricci_Flow__Chow

5. 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.