- SOME FURTHER CALCULATIONS RELATED TO :F AND W 279
If a is a 1-form, then the minimizer of the energy
E(h) ~JM ldh+al^2 dμ
is given by
/:::;.h = - div (a).Thus, for a steady gradient soliton g fl.owing along \7 f,
::2F(g, f) 2::: JM ( \liVjp - ~\lpVij) (\lpVij) e-f dμ
(6.110) -~ r ldivv-v(\7J)l^2 e-fdμ
2JM
+~JM ldivv - v ('VJ)+ \7wl^2 e-f dμ,
where
/:::;.w =-div (divv - v (\7 J))
and with equality in (6.110) if and only if(6.111) /:::;.(h-~) =-~div(divv-v('VJ)).
Since
we conclude
JM (div v - v (\7 J), \lw) dμ= - JM w div (div v - v (\7 f)) dμ= JM w/:::;.wdμ = - JM l'Vwl2
dμ,d2
ds2F(g, J) 2::: JM f( Y"iVjp - 1) 2\lpVij ('VpVij) dμ- lf 2 JM l'Vwl^2 dμ.Since
JM \liVjp \lpVijdμ = -JM Vjp \7i\7pVijdμ
=JM ldivvl
2
dμ +JM Vjp (-RpqVqj + RipjqViq) dμ,we obtain the following:
PROPOSITION 6.91. Let (Mn, g) be a steady gradient Ricci soliton flow-·ing along \7 f on a closed manifold. If ~~ = v and ~{ = h, then
(6.112) d2 F(g, J) 2::: { ( -! l'Vvl2 + ldivvl2 -! l'Vwl2 ) dμ,
ds^2 JM -RpqVqjVjp + RipjqViqVjp