1547671870-The_Ricci_Flow__Chow

30 2. SPECIAL AND LIMIT SOLUTIONS

and define
7/J(z )
p(r) = p(r(z)). ~ - r.

Then the metric may be written as

g = dr2 + r2p2 de2.

Introduce ' Cartesian coordinates'

x ~ rcose,

y ~ rsine.

Then r^2 = x^2 + y^2 and we have

whence

x

dx = -dr -yde

r

dy = ¥_ dr + x de,

r

x y

dr = - dx + -dy

r r

y x

de= --r2 dx+ -r2 dy.

A simple calculation shows that

g = ( 1 + p2 r~ 1 y2) dx2 - ( 2xy p2 r~ 1) dx dy + ( 1 + p2 r~ 1 x2) dy2.

So g extends to a smooth metric if

f (r) ~ p(r):-1

r

extends to an even function which is smooth at r: = 0. By Taylor's theorem,

f ( )

= p (0) - 1 2p (0) p' (0) ~ dk (p^2 ) rk-^2 ( K -1)

r r2 + r + L.t drk kl + 0 r.

k=2 r=O

Hence f yields a smooth even function at zero if and only if p (0) = 1,

p' (0) = 0, and all odd derivatives of p^2 vanish at the origin. Noting that

dk(2) k- l

__ P_ = 2pp(k) + '"'c ·kP(j) P(k- j)

drk L.t J

j=l

for appropriate integers Cjk, we conclude by induction that f yields a smooth

even function at zero if and only if p (0) = 1 and p(^2 k+l) (0) = 0 for all k ?: 0.

To translate back to 7/J (z) = rp (r), we simply observe that

dk7/J dk-l p dk p

drk = k drk-l + r drk.

In particular, the observation

1 d7/J d7/J dp

• --= - =p+r-
  <p d z dr dr'