498 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS
PROOF. We compute using the evolution equation for Eij that:t JM (detEt" dμ
=JM (detEt" (a ( (E-
1
)ij gtEij - 9ij (fftgij)) + C) dμ= a JM ( det E) a ( E-^1 ) ij \7 k \7 £ (Ek£ Eij - Eik Ej£) dμ
- JM ( det Et' (a ( ( E-^1 ) ij ( ~ det E ij - C Eij) + 2C) + C) dμ
=-a JM V'k [(detEt" (E-^1 )ij] ( Ek£\7gEij - Ej£\7gEik) dμ
+(1-2a) JM(detE)aCdμ.
Since'h [(detEt" (E-^1 )ij]
= (detEt" (a (E-^1 )pq (\7kEPq) (E-^1 )ij - (E-^1 )ip (E-^1 )jq \7kEpq),
we have
JM \i'k [(det E)a (E-^1 )ij] ( Ek£\7gEij - Ej£\i'gEik) dμ
=a JM (detE)u (E-^1 )pq \i'kEpq (Ek£ (E-^1 )ij \7gEij - \i'iEik) dμ
- JM (detE)u (E-^1 )ip (E-^1 )jq Y'kEpq ( Ek£\7gEij - Ej£\7gEik) dμ,
and the lemma follows from \i'iEik = 0. D
To see the monotonicity in the a= 1/2 case, we decompose the 3-tensor
Tijk into its irreducible components. In particular, the orthogonal group
0 (3) with respect to the metric E-^1 acts on the bundle of 3-tensors which
are symmetric in the last two components. The irreducible. decomposition
is given by
where the coefficients -i1 0 and ~ are chosen so that U is trace-free: