1547671870-The_Ricci_Flow__Chow

3. ANCIENT SOLUTIONS 29

FIGURE 1. Shrinking round n-sphere

3.2. The cylinder-to-sphere rule. Before considering our next ex-
ample, it will be helpful to recall the following result from Riemannian ge-
ometry.

LEMMA 2.10. Let 0 < w :::; oo, and let g be a metric on the topological

cylinder (-w, w) x sn of the form

g = <p (z)^2 dz^2 +1/J (z)^2 gcan,

where <p, 1jJ : (-w, w) -+ lR+ and gcan is the canonical round metric of radius

1 on sn. Then g extends to a smooth metric on sn+l if and only if

(2.15) 1-~ <p (() d( < oo,

(2.16) lim 1jJ (z) = 0,

z--+±w

(2.17)

. 1/J' ( z)

z--+±w lim -<p ( -) Z = +1,

and

(2.18)

. d2k1/J

z--+±w lim d S^2 k ( z) =^0

for all k E N, where ds is the element of arc length induced by <p.

PROOF. Since the result is standard, we will only prove sufficiency. The

argument is essentially two-dimensional, so to simplify the notation we shall

assume that n = 1 and

g = <p (z)^2 dz^2 +1jJ (z)^2 dB^2 ,

where e E JR/27rZ. It will suffice to consider the 'north pole' z = w. Let

s (z) ~ foz <p (() d(,

and set s ~ limz--+w s (z) < oo. Let

r(z)~s-s,