- THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 231
EXERCISE 6.13 (First variation formula for We:). Show that, on a closedmanifold Mn, if
(6.28) 5g = v, 5f = h, 5r = (
at (g, f, r), then
(6.29)r ( (-TVij + (9ij) sfj )
5(v,h,()Wc:(g, f, r) =JM + (~ _ h-~)(Ve+ e) udμ,where Sfj is defined by (6.27). In particular, ifthen
(6.30) 5(v,h,() Wc:(g, f, r) =JM (-TVij + (9ij) Sfjudμ.
SOLUTION TO EXERCISE 6.13. We compute5vRij = \7 P (5rfj) -\7i (5r~j)
and
5(v,h) (\7i\7jf) = \7i\7j (5f) - (5rfj) \7pf.
Adding these two formulas together, we get
5(v,h) (Rij + \7i\7jf)
= \7p (5rfj) -(5r~) \7pf +-vi (-vj (5f) - 5r~j)
(6.31) =ef\7p(e-f5rfj)+\7i\7j(h-~)·
Tracing this formula, remembering to take the variation of gij, and using
gs T = (imply
5(v,h,() [r (R +fl.!) (47rr)-~ e-f dμ]
= T (-5gij · (Rij + \7i\7jf) + ij · 5 (Rij + \7i\7jf)) (47rr)-~ e-f dμ
+r(R+i:l.f)(47rr)-~ ((1-~)~-h+ ~)e-fdμ= T (47r7) 2 dμ.
[-Vij (Rij + \7i\7jf) e-f + \7p (e-f gsr~) j _13:
+e-f fl. (h-~) + (R+ !:l.f) e-f (~ - h-n;-^2 ~)