- ENERGY, ITS FIRST·VARIATION, AND THE GRADIENT FLOW 193
So using (5.8),6 [(R+L1f)e-fdμ]= gij5 [(Rij + "Vi"Vjf) e-f dμJ - 6gij · (Rij + "Vi"Vjf) e-f dμ
= ["Vp(e-fgijorfj)+e-fl1(h-~)+(R+L1f)e-f(~ -h)Jdμ
- Vij · (Rij + "Vi"Vjf) e-f dμ.
Note that ori is a tensor and we do not need an explicit formula for it in
the rest of the proof.
By the Divergence Theorem, we have
b(v,h)F(g,f) =JM 6 [(R+L1f)e-fdμ]
= JM(-L1(e-!)+(R+L1f)e-f) (~ -h)dμ
- JM Vij · (Rij + "Vi"Vjf) e-f dμ,
from which the lemma follows. D
REMARK 5.4. By (5.11), the variation of (Rij + "Vi"Vjf) e-f dμ is a di-
vergence when h = ~:
6 [ ( Rij + '\7 i '\7 j f) e-f dμ] = '\7 P ( e-f orfj) dμ.
Note also the factor ~-h in front of the second term in the RHS of (5.10).
The significance of when this factor vanishes will be seen in subsection 1.4
below. By (5.8) we have
LEMMA 5.5. Define the measure
dm ~ e-f dμ.If the variations of g and f keep the measure dm fixed, that is, b(v,h) (dm) =
0, then
(5.12) V=2h.
As a consequence of Lemma 5.3, we have
COROLLARY 5.6 (Measure-preserving first variation of F). For varia-
tions (v, h) with 6(v,h) (e-f dμ) = 0, we have
(5.13) b(v,h)F (g, J) = - JM Vij (Rij + "Vi"Vjf) e-f dμ.
Notice in formula (5.10) for b(v,h)F (g, f) the occurrence of the terms(5.14) Rij ~ (Rcm)ij ~ Rij + "Vi"Vjf,
(5.15) Rm ~ R + 2L1J - ['\7 f [^2.