1547671870-The_Ricci_Flow__Chow

58 2. SPECIAL AND LIMIT SOLUTIONS

In this way, one obtains part (3) of Proposition 2.36 as an immediate corol-
lary of Lemma 2.40.

5.5. Neckpinches happen. In this part of the analysis, we show that
there exist initial data '1i = 1jJ (0) meeting our hypotheses. In particular,
we construct simple examples obtained by removing a neighborhood of the
equator of a standard sphere and replacing it with a long thin neck. These
examples satisfy '1i = .J A+ Bs^2 near the equator (for appropriate constants
A and B) and blend smoothly into the standard sphere metric on the polar
caps. Our construction justifies

PROPOSITION 2.42. There exist initial metrics

g = ds2 + w2 9can

for the Ricci flow on sn+l which satisfy I '1i s I ::::; 1, have positive scalar cur-

vature, and possess a neck sufficiently small and a bump sufficiently large

so that under the flow, the neck must disappear before the bump can vanish.

Hence these solutions exhibit a neckpinch singularity in finite time.

Our method will be to remove a sufficiently large neighborhood of the

equator of the round metric ds^2 + (cos s )^2 9can on sn+l and replace with a

sufficiently narrow neck. Let A > 0, and let B satisfy

0 < B < 1/2 if n = 2

0 < B < 1 if n 2": 3

Define the function

W(s) ~ JA+Bs2.

LEMMA 2.43. The metric

ds^2 + W (s)^2 9can

on JR x sn has positive scalar curvature, and the scale-invariant measure of

its curvature pinching

a= W(s)W"(s) - W'(s)^2 + 1

obeys the bounds

1-B <a< 1 +B.

PROOF. One computes that

W(s)^2

--· R = -2W(s)W"(s) + (n - 1){1 - W'(s)^2 }

n

B2s2

= n - 1 - 2B + (3 - n)--W(s)^2.

Since Bs^2 <A+ Bs^2 = W(s)^2 , we find that when n 2": 3,

W(s)^2

--· R 2": n - 1 - 2B + 3 - n = 2(1 - B).

n