496 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWSREMARK B.22. The above proof follows [34], where it is pointed out
that the proof of short-time existence in [106] is incomplete.3.3. Monotonicity formulas for the XCF. Given short-time exis-
tence of the flow, one would like to prove long-time existence and convergence
of the flow on closed 3-manifolds. Although this problem is still open, we
have the following result due to Hamilton, from [106], which is Proposition
3.24 on p. 88 of Volume One.
PROPOSITION B.23 (Monotonicity formulae). If (M^3 , g (t)) is a solution
to the cross curvature flow with negative sectional curvature on a closed 3-
manifold, then
(1) (volume of Einstein tensor increases)(B.38)8
ot Vol (E) ~ 0,(2) (integral difference from hyperbolic decreases)(B.39) d f (1 .. (detE)
1
/3- -(l^3 Ei") - -- ) dμ :S 0.
dt M3 3 J detg
REMARK B.24. Applying the arithmetic-geometric mean inequality to
the eigenvalues of Eij with respect to gij, we see that the integrand in part
(2) is nonnegative, and it is identically zero if and only if Eij = ~Egij, that
is, if and only if gij has constant sectional curvature.
In the remainder of this section, we prove the above two monotonicity
formulas. We begin by computing the evolution of the Einstein tensor.
LEMMA B.25.:tEij = \lk Ve ( Ek£Eij - Eik Ej£) - det E gij - C Eij,
where C ~ gij Cij.PROOF. Taking q = j in (B.31), we may rewrite the Einstein tensor as
(B.40) Emn _ - -4^1 μijmμk£nR ijk£·
From (B.40) and (B.34) with Vij = 2Cij, we compute
8tE^8 mn = -4μ^1 ijm μ k£n ( \7i\7£Cjk + Vj\7kCi£ - \7i\7kCj£ - \7j\7£Cik )
- lμijmμk£ngpq (RijkpCq£ +-Rijp£Cqk) - 2CEmn