- A GEOMETRY WITH ISOTROPY SO (2)
there is by (1.6) some k3 < oo such that
!!:F <PF= ±_AF< k^3
dt - BC - k2 (k1 - 8t).
Hence there is a positive function Ji (t) bounded on [O, T] such that
F (t , ·) :S Ji (t)
for all t E [O, T]. Then since
!!:_A< 4F^2 < 4 (! 1 (t))^2
dt - BC - k2 (k1 - 8t)'there are positive functions h (t) and f3 (t) bounded on [O, T] such that
A (t, ·) :Sh (t)
and thus
p 4h (t)
(t, ·) ::::: k2 (k1 - 8t) ::::: h (t)
for all t E [O, Tj. Finally, since E^2 /(BC) 2:: 4, we have
16 16
Q (t , ·) 2:: A 2:: h (t)
for all t E [O, Tj. Hence as E '\, 0, we have
EP - Q ---7 -oo
Euniformly on [O, Tj, which proves the result.
6. The Ricci flow of a geometry with isotropy SO (2)
150
Under the Ricci flow, the 52 factor of 52 xJR shrinks homothetically while
the JR factor remains unchanged. As a consequence, the solution becomes
singular in finite time, and the manifold converges in the pointed Gromov-
Hausdorff sense to JR. In Section 5 of Chapter 2, we will discuss Ricci
fl.ow singularities on portions of a manifold that are geometrically close to
5n-l x (-€, €).
On the other hand, the product H^2 x JR and the Lie groups nil and
SL ----------(2, JR) all share the property that certain directions selected by the Lie
algebra expand under the Ricci flow while the remaining directions converge
to a finite (possibly zero) value. As a result, solutions of the normalized Ricci
flow on compact representatives of the latter three geometries all collapse
with bounded curvature, exhibiting Gromov- Hausdorff convergence to 1-or
2-dimensional manifolds. (See [76].)
We shall use nil as an example of a geometry with isotropy SO (2),