1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

12. THE HEAT-TYPE EQUATION SATISFIED BY Q 115

where Z = ~ \7 6. 5 2v + idv. Now

1 2.

3tr' (\7 S(Z 0 g52))jk = \7' (Zi(gs2)jk + Zj(gs2)ik + Zk(gs2)ij)

= div(Z)(gs2)jk + \7kZj + 'VJZk.

Therefore

(29.159)

tr^1 '^2 (\7TF(B)) = \7^2 6. 5 2v + l

3

0\7^2 v - ~6. 5 2vg 52 - div(Z)g52 - 2S(\7Z)

= 2 1( \7^2 6.52v -^1 2 ) (^2 1 )

2

6. 5 2vg 52 + 3 \7 v -
   2
   6. 5 2vg 52.

We conclude that a in (29.155) satisfies

(29.160) tr^12 ' (a)= 2 v ( \7^2 6.52v - 26.^1 52 2vg52 ) + 3v ( \7^2 v -^1 )

2

6.s2vg52

• 2tr^1 '^2 (dv 0 TF(B)).

Second, for the second line of (29.154), we shall derive

(29.161) JVvJ^2 JTF(B)J^2 = 2 ftr^1 '^2 (dv 0 TF(B))f

2

.

Given any point p E S^2 , we shall compute in local coordinates where 9iJ = Oij at

p. We have

(29.162a)

(29.162b)

This implies that

2

(29.163) JTF(B)J^2 = L (TF(B)iJk)^2 = 4((TF(B)m)^2 + (TF(Bb2)^2 ).

i ,j,k=l

Using (29.162) and (29.163), we calculate

ftr^1 '^2 (dv 0 TF(B))f

2

= (\71vTF(B)m)

2

• (\71vTF(B)122)

2

+ 2 (\71 v TF(B)112)^2 + (\72v TF(B)211)^2

+ (\72v TF(B)222)^2 + 2 (\72v TF(B)212)

2

= 2 (\71 v)

2

(TF(B)m)

2

• 2 (\71 v)

2

(TF(Bb2)

2

• 2 (\72v)^2 (TF(Bh22)^2 + 2 (\72v)^2 (TF(B)m)^2

= ~ J'Vv J

2

JTF(B)J

2

.

2

This establishes (29.161).