1547671870-The_Ricci_Flow__Chow

5. THE NECKPINCH 57

Now we integrate once again to get

1

'1/J du

v?J(J" >.

• Tmin. /logrmin - 1

V logu

Substitute u = rminV· Then it follows from the inequalities rmin ::::; u ::::; 'I/;

that 1::::; v::::; 'l/;/rmin· By (2.64), this implies that 0::::; logv::::; -~~ logrmin·
Thus we have

(2.65a) --> mm dv

JC(}" f,'!/J/rmin v-logr. - logv

r min - 1 JlDg"V

(2.65b)

1 f,'!/J/rmin dv

~ - 2 J-logrmin ~·

1 v logv

To put this inequality into a more useful form, we shall use the following
fact, which is left to the reader to verify.

CLAIM 2.41. The function Z: [O, oo) ~ [1, oo) defined by

~ = r z ( ~) ____!!:!.___

11 vrogv

is monotone increasing and has the asymptotic behavior that

1

z (~) = 1+2e+0 (e) as ~ ~ 0

and

Z(~) = [1 +o(l)]~~ as ~ / 00.

In light of the claim, estimate (2.65) can be recast as

'!!!._ < z ( 2vc(J" ).

r - rminv-logrmin

Estimate (2.61) follows from this and the expansion of Z(~) for small~·

For larger O" we get

'l/J (}" (}"

- :::;c log ,

rmin rminv-logrmin rminv- logrmin

which is exactly (2.62). This estimate will be valid in the region where

(2.64) is satisfied and O" ~ b'rminv-logrmin· Using Cv-logrmin = r~i~,

we conclude that (2.62) will hold if 'l/;/rmin::::; (1/rmin)c:^2 /^2 C+o(l). 0

By Lemma 2.37, the neckpinch forms a Type I (rapidly forming) singu-

larity. One can therefore construct a sequence of parabolic dilations near

the developing neck which converge to a singularity model that is an ancient

solution of the Ricci fl.ow. (See Subsection 4.1 of Chapter 8.) The lemma

above shows that the singularity model must in fact be the cylinder solution

(2.41). It follows that

rmin (t) = [1 + o (1)] J2 (n - 1) (T - t).