- THE ENTROPY FUNCTIONAL W AND ITS MONOTONICITY 233
EXERCISE 6.15 (Evolution of v 6 and monotonicity of W 6 ). Consider the
gauge transformed version of (6.32)-(6.34) on a closed manifold Mn:^9
8(6.35) at9ij = -2Rij,
of 2 nE:
(6.36) at = -b..f - R +IV fl -
27,
dr
(6.37) dt = E:.Let v 6 ~ Vcu. Show that if (g ( t) , f ( t) , T ( t)) is a solution of the system
above, then(6.38)Also show that this implies
! W 6 (g (t), f (t), T (t)) = 2r JM IRij + \7i\7jf + 2 E:
79ijl2
udμ= 2r JM iSfj \
2
udμ 2:: 0.The next exercise relates the first variation of W 6 to the linear trace
Harnack quantity.
EXERCISE 6.16 (Variation of W 6 and linear trace Harnack). Show that,for any symmetric 2-tensor won a closed manifold Mn, we have:
(6.39) JM WijSfjudμ =JM Z (w, \7 f) udμ,
where Z is the linear trace Harnack inequality defined in (A.27)
E:
(6.40) Z (w,X) ~ 'Vj'ViWij + RijWij +
27
W - 2\liWijXj + WijXiXj
and W ~ gijWij. In particular, if b(v,h,() ( ( 47rT )-n/^2 c f dμ) = 0, then
b(v,h,()WE:(g,f,r) =JM Z(w, \lf)udμ,
where
Wij ~ -TVij + (gij = -T^2 b(v,() (r-^1 g).
HINT. Use the identityJM 'Vj'ViWije-f dμ =JM \liWij'Vjfe-f dμ
=JM Wij (\7if\7jf - \7i\7jf) e-f dμ.
In the last two exercises in this subsection, we first rewrite W 6 (g, f, r)
and then we use the new formula to give a lower bound for W 6 (g, f, r).
(^9) These equations are equivalent to (6.35)-(6.37) after pulling back by diffeomorphisms
generated by the vector fields '\J f (t).