1547671870-The_Ricci_Flow__Chow

4. UNIQUENESS OF RICCI SOLITONS 119

where K = R/2 denotes the Gauss curvature of g.
We now derive the form of (5.17) that we used in proving Proposition

5. 
   
   
   
   21. The real vector space of conformal Killing vector fields of ( S^2 , g) is
       6-dimensional. Three of the dimensions arise from vector fields of the form
       V cp, where cp is a spherical harmonic. The other three dimensions come from
       the Killing vector fields for g. We claim that if Y is a Killing vector field for
       g, then
   
   
   
   

f Y(K) e^2 udA=0.

Js2

By the dimension count 6 = 3 + 3, the claim implies that for any conformal

vector field X, one has

(5.18) r (X, VK) dA = r x (K) dA = 0,

Js2 Js2

since dA = e^2 u dA. Note that conformality is well defined, because X is

conformal with respect to g if and only if X is conformal with respect to g.

The claim is derived from an integration by parts. Since R = 1, Lemma

5.3 implies that

K = e -^2 u ( 1 - 3. u).

Recalling that div Y = 0 whenever Y is a Killing vector field for g, we

compute that

f Y(K) e^2 udA= f (Y,VK 9 )_ e^2 udA

Js2 Js2 g

1

= - 2 K \ / -Y, \l u ) _ e 2u dA. -

52 g

Now

f K (Y, Vu)_ e^2 u dA = f (1 - 3.u) (Y, Vu)_ dA

Js2 g Js2 g

= - f (3.u) (Y, Vu)_ dA,

Js2 g

because

f (Y, Vu)_ dA = - f u. divY dA = o.

Js2 g Js2

Integrating by parts again and using the fact that VY is antisymmetric, we

obtain

f (3.u) (Y, Vu)_ dA = - f ViuViVjuYj dA - f ViuVjuviyj dA

~2 g ~2 ~2

= -~ 2 }f (Y, V \vu\:)_ dA

52 g^9

= ~ 2 }f \Vu\: (divY) dA = o.

52 g