- THE CROSS CURVATURE FLOW
Since this decomposition is orthogonal with respect to E-^1 , we have
(B.42) lrijk -Tjik,2 = juijk - ujik,2 + 1-~EikTj + ~Ejkri/2
E-1 E-^1 2 2 E-1(B.43) = juijk - ujikl:-1 + ITil~-1.
Taking a= 1/2 in the lemma, we conclude that
!_ { (detE)^112 dμ = ~ { juijk Ujik,
2
(detE)^112 dμ 2: O
dt JM 4 JM E-1
and part (1) of Proposition B.23 follows.
We now give the proof of part (2) of Proposition B.23. DefineJ-:::;:. JM ( gij :ij - ( det E)l/3) dμ,
where 9ijEij = -!R. We compute
dt d JM r (.. ) g2J Eij dμ = JM r [(a at9ij ) EiJ .. + 9ij at a EiJ .. + (lJ .. Eij) c ] dμ
499=JM (2cijEij - 9ij ( det E gij + C Eij) + (gij Eij) C) dμ
= 3 JM detEdμ.
Combining this equation with (B.41) for a= 1/3 and (B.43), we conclude
dJ = -~ r (~ jrijk -Tjikl2 - ~ ITi\2 ) (detE)l/3 dμ
dt 3 JM 2 E-1 3 E-^1+JM detEdμ-~JM (detE)
1
/3
Cdμ=-~JM (juijk - ujikl:-1 + ~ \Ti~-1) (detE)1/3 dμ
- JM(~ -(detc)^113 ) (detE)^1 /^3 dμ
:::; 0,where we used detE = (detc)^113 (detE)^113.
3.4. Cross curvature solitons. A solution to the XCF is a cross
curvature soliton if there exists a vector field V and .\ E IR such that at
some time
(B.44)More generally, a solution to the XCF is a cross curvature breather if
there exist times ti < t2, a diffeomorphism cp : M ----+ M, and a > 0 such