1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

34 17. ENTROPY, μ-INVARIANT, AND FINITE TIME SINGULARITIES

where

D*=-~-~+R

· at

and where (l\7f Rm) (X, Y, Z) =Rm (\7 J, X, Y, Z).

If g 8 g = v and fsJ =~'then

(17.106)

where the subscript (1, 4) denotes the components on which the inner prod-

uct is acting. As a special case, if Rc+\7\7 f::::::: 0, then ifs J 1Rml^2 e-f dμ =

-~ J (v, a) e-f dμ.

We summarize the above formulas. For a = 1, 2, define the 2-tensors
13(a), the functions /(a), and the functionals Fa (g, J) by

(17.107a)

(17.107b)

(17.107c)

(17.107d)

(17.107e)

(17.107f)

13(^1 ) = -2 (Rc+V'V'f),

13(^2 ) =-a,

1(^1 ) = -R - ~f,

1(2) = _! IRml2'

2

Fi(g,J) =JM (R+ IY'fl

2

) e-f dμ,

F2 (g, f) = 1JM1Rml^2 e-f dμ.

Given a functional g (g, J), let 6(f3(a) ,f'(a) )g (g, f) denote Js g (g (s), f ( s))

under g 8 g (s) = 13(a) and gj (s) =/(a)_ We have

(17.108)

and

(17.109)

6(/3<1J,/'(1J) (11Rml

2

e-f dμ)

= -1 div ( e-f'V 1Rm1^2 ) dμ + L (a, \7 J) e-f dμ

-IP* -l\7JRml^2 e-fdμ

6(/3<2J,f'(2J) ( ( R + 2~f - IV' Jl^2 ) e-f dμ) = -L (a, \7 J) e-f dμ

+ 2 (a, Rc+\7\7 J) e-f dμ,