- SOME FURTHER CALCULATIONS RELATED TO :F AND W 273
Applying the Holder inequality,JM IY'fl4udμ 2 (JM IY'fl^2 udμ)
2
= 16,\i,which we substitute into (6.104) to conclude
,\ n(n-1)
1:::; 4.- Some further calculations related to F and W
DIn this section we discuss some interesting computations related to en-
ergy and entropy including variational formulas for the modified scalar cur-
vature, the second variation of energy and entropy, and a matrix Harnack
calculation for the adjoint heat equation.
7.1. Variational structure of the modified scalar curvature.
7.1.1. Variation of the modified scalar curvature. We now give yet an-
other proof of the variation formula for the F functional (5.10) when of~
h = ~ using the pointwise formula for the variation of the modified scalar
curvature. The formulas for Vo ~ R + 2b..f - IV' fl^2 and F below should
generalize to Vs and W 6 (see (6.25) and (6.26) for the definitions of Vs and
Ws)·LEMMA 6.82 (Measure-preserving variations of R + 2b..f - IV' f 12 and
the linear trace Harnack). If og = v and of = ~' where V ~ gijVij, on a
manifold Mn, then
(6.105)
o(v,~) ( R + 2b..f - IV' fl
2
)
= V'iY'jVij + VijRij - 2\i'iVik Y'kf + VijY'ifV'jf - 2Vij (Rij + Y'i Y'jf) ·For the proof see the more general Lemma 6.85 below.
Since under our assumptions o (e-f dμ) = 0, we have, using (6.105) and
the identity (6.39) with c = 0, that
a;:.= JM (o(v,~) (R+2b..f-IY'fl
2
)) e-f dμ= - JM Vij (Rij + Y'iY'jf) e-f dμ.
This is the special case of (5.10) where h = ~-
REMARK 6.83. If (Mn,g(t)), t E (-00,00), is a gradient Ricci soliton
fl.owing along V' f so that Rij + Y' i Y' j f = 0 and if