500 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS
that
(B.45)
We have the following nonexistence result.
LEMMA B.27 (XCF breathers are trivial). If (M^3 ,g(t)) is a cross cur-
vature breather with negative sectional curvature, then g (t) has constant
sectional curvature.
PROOF. We have
d r ..
dt Vol(g(t)) = }Mg2^2 cijdμ > 0,so that the breather equation (B.45), i.e., g (t2) = acp*g (t1) for ti < t2,
implies a> 1. On the other hand, J(g(t2)) = a^112 J(g(t1)) 2:: 0, which
contradicts the monotonicity formula (B.39) unless J (g (t2)) = J (g (t1)) =
0, in which case g (t) has constant sectional curvature. 0
4. Notes and commentary
Lemma B. 7 is Lemma 3.3, Theorem 3.4 and Corollary 3.5(i) in [212].
See Ma and Chen [259] for a study of the cross curvature flow for certain
classes of metrics on sphere and torus bundles over the circle. For work on
the stability of the cross curvature flow at a hyperbolic metric, see Young
and one of the authors [235].