192 5. ENERGY, MONOTONICITY, AND BREATHERSdefine V ~ gij Vij. Routine calculations give(5.6) ovrij k (g) = 2,g^1 kl ('\liVjl + 'Vjvil - '\lzVij ) ,
1(5.7) ovr~j = 2VjV,
(5.8) O(v,h) ( e-f dμ) = (~ - h) e-f dμ.
We calculate the last one, for example,(5.9) O(v,h) (e-f dμ) = -e-fh dμ + e-f ~gijVij dμ = (~ ·_ h) e-f dμ.
LEMMA 5.3 (First variation of F). Then the first variation of F can
be expressed as(5.10) O(v h)F (g, f) = - r Vij(Rij + '\li'Vjf)e-f dμ
' JM+JM(~ -h) (2L).f -1Vfl
2
+ R) e-f dμ,where O(v,h)F (g, f) denotes the variation of F at (g, f) in the direction
( v, h) , i.e.,O(v,h)F (g, f) ~ dd 8 I F (g + sv, f +sh).
s=O
PROOF. Recall (Vl-p. 92), i.e.,Rij = R~ij = aprfj - air~j + r{jr~q -r~jrfq,
so that
o Rij = v p ( orfj) -vi ( or~j).
Since Vi'Vj = aiaj - rfj8k as an operator acting on functions, we have
o (vivjf) = vivj (of) - (orfj) Vpf.
Hence, using (5.7),
o (Rij + vivjf) = Vp (orfj) - (orfj) Vpf +vi (vj (of) - or~j)
= ef'\lp (e-f orfj) + Vi'Vj (h-~).
We then compute
(5.11) o [(Rij + '\li'Vjf) e-f dμ]
= [ Vp (e-forfj) +e-fViV~(h-¥) l dμ.
+ (Rij + '\li'Vjf) e-f ( 2 - h)