220 5. ENERGY, MONOTONICITY, AND BREATHERS
using ~ = h ~ 8 f. Integrating by parts yields
8:F = 8 JM Re-f dμ + 5 JM IVJl^2 e-f dμ = - JM Vij (Rij + \7i"'\ljf) e-f dμ.
Although the Ricci tensor is not strictly elliptic in g, one can ask if the
RHS of equation (5.27)
.§_g· at tJ · = -2 [R tJ · + "'\7 i ·"'\7 J ·log ( dm dμ)]
is elliptic in g. The answer is still no. In particular, if 8g = v, then
Hence
Since
we have
8 (-2 [Rij + "'\li"'\ljlog (:~) ]) = b..vij - "'\li"'\lkvjk - "'\lj"'\lkvik
+. lower-order terms,
where the last pair of terms form a Lie derivative of the metric term. How-
ever the second-order operator on the RHS is still not elliptic in v.