1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

114 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

Applying (29.148), (29.149), (29.152), (29.153), andRv = (vl:. 52 v-IV'vl^2 +2v^2 )

to (29.147) and completing a square, we obtain

(29.154) ( :t -vl:.52) Q = -2 lvV'TF(B) + 2dv © TF(B)l^2 - 4Rv ITF(B)l^2

+ (6vl:.52v + 3 IV'vl

2

) ITF(B)l

2

+ 36vTF(B) ·TFS (dv © V'^2 v)

+ 6v TF( B) · (V' l:. 52 v © V'^2 v)

-12vTF(B) ·TFS (V'^3 v 83, 2 V'^2 v)

+ 6v tr^1 '^2 (dv © TF(B)) · ( V'^2 l:. 5 2v - ~l:.~2Vg5 2 ).

STEP 2. To evaluate the norm of

(29.155) a~ v \i' TF(B) + 2dv © TF(B),

which occurs in the first term on the RHS of (29.154), we need the following. Given

a 4-tensor c = L,;,j,k,£=l ceijkdxe © dxi © d x J © dxk which is both symmetric and

trace-free in the last three slots, an orthogonal decomposition for c is given by

(29.156)

where

e ·k =. 1 -0 £i Cn· ·k =^1 - tr^12 ' (c) ·k

J. 3 <iJ 3 J.

The tensors TF(c) and e are totally trace-free. We compute that

(29.157) lcl^2 - I TF(c)l^2 = 9 le l^2 =I tr^1 '^2 (c)l^2.

In particular,

(29.158)

First, we compute tr^1 '^2 (a). We h ave

( tr1,2 (V' B)) jk = (V' S(V'3v)) iijk

1 ( 4 4 4

= 3 \i'iijkv + \i'ijikv + \i'ikijv)

4 1 52 2 1 52 2

= \i'iijkv - 3Rjike\i'ieV - 3Rkij£\i'ieV

2 2 2 2

= l:.52\i'jkv - 3\i'jkv + 3l:.52v(g52)jk·

Now applying (29.142) to this yields

tr1 2 ( ' \i'B ) = V'^2 l:.^10 2 4

5 2v + -V' 3 v - 3 -l:. 5 2vg 5 2.

Since by (29.131a) we h ave that B = TF(B) + 3 S(Z © g 52 ) , it follows that

tr^1 '^2 (V' TF(B)) = tr^1 '^2 (V' B) - 3 tr^1 '^2 (V' S(Z © g 52 )),