- POSITIVE SECTIONAL CURVATURE DOMINATES 257
study of (6.32), we first consider the subcase that μ = v < 0. We then have
the following system of two ODE:
d>.. = >..2 + v2
dt
dv 2
dt = v + >..v.For convenience, we set 7r = -v > 0 and consider the system
d>.. = >..2 + 7r2
dt- d7r =AK - 7r^2
dt
under the assumption that >.. > v = -Ir initially. Since
d
dt ( >.. + 7r) = >.. ( >.. + 7r) 'this condition is preserved. Similarly, 7r > 0 as long as the solution exists.
If>..:::; 0, then d7r/dt:::; -Ir^2 < 0. If 7r 2 >..,then d7r/dt = 7r (>.. - 7r):::; 0
but d>../dt > 0. Hence 7r is non-increasing except possibly when 7r < >..,
and it follows that 7r :::; max {A, C}. To improve on this estimate, note that
d>../dt > 0, so that we may write
d7r
d>..
This is a homogeneous equation. The standard substitution 7r = >..( gives
d( ( - (2
( + >.. d>.. = 1 + ( 2.Using partial fractions and integrating, we obtain
J d: + J ( :2 -z + 1! () d( = 0,
which yields
1
log l>..1-( - log 1 (1+2log ll +(I= C.
Since ( = 7r / >.. and >.. + 7r > 0, we may write this in terms of >.. and 7r as
log 7r = ~ + 2 log ( 7r : >..) + C.If 7r (t) is sufficiently large and positive, this leads to a contradiction unless
>.. (t) > 7r (t) log Ir (t).
In particular, >.. is positive and much larger than Ir. On the other hand, if 7ris sufficiently small and positive, we get a contradiction unless -Ir < >.. < 0.
This discussion motivates the following theorem, which reveals the pre-
cise sense in which all sectional curvatures of a complete 3-manifold evolv-
ing by the Ricci flow are dominated by the positive sectional curvatures.