- ROTATIONALLY SYMMETRIC EXPANDING SOLITONS 29
This establishes that X < a always along these trajectories, and the equa-
tion
.· dX/ds = aY^2 + >.y'n"W^2 when X = O
shows that X remains positive.. Let L = X^2 + Y^2. Then
dL/ds = 2(L-l)(X^2 - .AW^2 ) + 2.Ay'n"W^2 (X - a)
shows that L :S 1 is preserved. Finally, when X <a,
dW/ds :S W(a^2 _:_ >.W^2 ),showing that W cannot exceed a/./>..= 1/~. D
LEMMA 1.40. As s ---> +oo, these trajectories approach the origin in
W XY coordinates,.. and the corresponding metrics are complete.PROOF. Because the right-hand sides of the system (1.60) are polyno-
mial, parameter s is unbounded along these trajectories. Because the trajec-
tories are bounded within the region given by Lemma 1.39, each trajectory
must limit to either the critical point H or to the origin 0 as s ---> +oo.
However, linearization shows that the tangent space of the stable manifold
of H intersects the closure of the region given in Lemma 1.39 only in the line
through H tangent to the hyperbolic trajectory. Thus, all the trajectories
under consideration limit to the origin..
To show that the metrics corresponding to these trajectories are com-
plete, note that
(1.61)Along these trajectories, W must eventually become a decreasing function
of s. Then it suffices to show that there is a point (Wo, Xo, Yo) along one of
our trajectories such that the limit ·
1
Wo dWl~ e .AW2-x2
is infinity; this follows from .>. W^2 - X^2 < .>. W^2. D
The flat trajectory divides our chosen quadrant of the unstable manifold
near S into regions of negative sectional curvature (bordering the hyperbolic
trajectory) and regions of positive sectional curvature (bordering the steady
soliton trajectory). The following lemma shows ·that these signs· persist;
thus, there exist 1-parameter families of rotationally symmetric expanding
solitons of strictly positive and of strictly negative sectional curvature.
PROPOSITION 1.41. Excluding the fiat metric, the sectional curvatures
of these metrics are either strictly positive or strictly negative for all r.