1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

224 6. ENTROPY AND NO LOCAL COLLAPSING

Combining the above three formulas and simplifying a little, we get

b(v,h,() W (g, f, r)

= - JM rVij(Rij + '\li'\ljf)udμ

+/Mr(~ -h) (2Llf-1'7fl

2

+R+f~n)udμ

+JM [h+ (1-~) ((R+ l'Vfl

2

) - ;;(f-n)] udμ.

We rewrite the above expression as

b(v,h,() W (g, f, r)

=JM (-rVij + (gij) (~j + '\Ji'\Jjf)udμ

• JM r ( ~ - h - ;; ) (2Llf -I 'V f I 2 + R + f ~ n) udμ

+JM -((R+L'.lf)udμ+ JM~( (2Llf-1'7fl

2

+R)udμ

+JM ( 1-~) ((R+ l'Vfl^2 )udμ +JM hudμ,

which, by combining terms, we further simplify to

b(v,h,() W (g, f, r)

=JM (-rVij + (gij) (~j + '\Ji'\Jjf)udμ

+]Mr(~ -h-;;) (2Llf-1'7fl

2

+R+f~n)udμ

+JM (n-1) ( ( Llf-1'7!1

2

) udμ +JM hudμ.

Since (is a constant, (6.9) follows from a rearrangement and the integration

by parts identity: JM ( Llf - l'V f1^2 ) e-f dμ = 0. D

REMARK 6.2. Analogous to (5.14) and (5.15), the terms

1 2 f-n

Rij + '\li'Vjf - -

2

9ij and R + 2f:.:.f - l'Vfl + --

r r

in (6.9) are natural quantities vanishing/constant on shrinking gradient Ricci
solitons.

1.1.3. The gradient flow of W. When we require that the variation

( v, h, () satisfies

V n

( = -1 and

2

• h -
  27

( = O,