- THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 223
1.1.2. The first variation of W. Let Og = v E C^2 (M, T M @T M), let
of= h, and let OT=(. Since
(6.7) o (udμ) = o ( (47rT)-nl^2 e-f dμ) = (-;:_ ( - h + ~) udμ,
where V ~ trg v = gijvij, the measure (47rT)-nl^2 e-f dμ preserving variations
satisfy(6.8)n V
--(-h+-=0.
2T 2
We find it convenient to write the variation of W so that this quantity is
one of the factors. In particular, we haveLEMMA 6.1 (Entropy first variation formula). The first variation of W
at (g, f, T) can be expressed as follows:OW(v,h,() (g,f,T)(6.9) =JM (-TVij + (9ij) (Rij + \h\Jjf -
2~9ij )udμ+!MT(~ -h-;~) (R+2~f-l\7fl2
+f-~-l)udμ.PROOF. It follows from the first variation formu,la (5.10) of :F with re-
spect tog and f (keeping T fixed) that
O(v,h,O) (T(47rT)-nf^2 :F(g,f))= - JM TVij(Rij + \7i\7jf)udμ
+!MT(~ -h) (2~f-l\7fl
2
+R)udμ.Next we calculate the first variation of the remaining term of W with respectto g and f (again keeping T fixed),
O(v,h,O) ((47rT)-n/^2 JM(!-n) e-f dμ)
=JM [(l+n-f)h+ ~(f-n)] udμ.
Now the term from the variation of W with respect to T is
· O(o,O,() (JM [T(R + l\7 fl^2 ) + f - n] (47rT)-nl^2 e-f dμ)
= JM [ ( 1 - ~) ((R + l\7f1
2