1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

130 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

The proof of this lemma shall occupy the rest of this section. The totally

symmetric and trace-free 3-tensor a, defined by (29.171) in Euclidean coordinates,

is given invariantly by a= \7^3 v-~ S(6euc 'Vv®geuc), where \7 denotes the Euclidean

covariant derivative. Let (r , B) be polar coordinates. By (29.175), now using the

orthonormal frame 8 ~ 1 = gr and 8 ~ 2 = ~ %e, we have on ffi.^2 that

where

et=;= -. n3 VrrrV--^3 r -2n3 veerV->

(3-:::;=. r - 3 nveeeV^3 - -^3 r -ln2 vrrev. -

Next, we express a and /3 in terms of the partial derivatives of v with respect

to r and (}. We have I'~e = I'~r = ~ and f&e = -r and the rest of the Euclidean

Christoffel symbols are zero. Therefore the components of the Hessian of v are

given by

n2 v rrV - = Vrr, -

'V~ev = Vee + rvri

1

v;ev = Ver - -ve.

r

Taking another covariant derivative, we have

Therefore,

n3 v rrrV - == Vrrr, -

'V~eev = veee + 3rvre - 2ve,

2

'V~erv = Vree - -Vee+ rvrr - Vr,

r

2 2

v:rev = Verr - -Ver + 2Ve.

r r

a= Vrrr - 3r-^2 vree + 6r-^3 vee - 3r-^1 vrr + 3r-^2 vr ,

/3 = r-^3 veee - 3r-^1 verr + 9r-^2 vre - 8r-^3 ve.

To express Q o u-^1 in terms of the partial derivatives of v with respect tor and

e, we first express Q in terms of the partial derivatives of v with respect to 'lj; and

B. Since 8 ~1 = f..p and 8 ~2 = sec 'lj; %e are orthonormal with respect to g 5 2, from

(29.163) and (29.169), we have on S^2 that

Q = v ITF(B)l

2

= ~ (a^2 + (3^2 ),

where

a~ \7~..p..p v - 3 sec^2 'lj;'V~eev - 2\7..pv,

f3 ~ sec^3 'l/;'V~eev - 3 sec 'lj;\7~..p..pv - 2 sec 'lj;\7 ev.