480 B. OTHER ASPECTS OF RICCI FLOW AND RELATED FLOWS
In higher dimensions, there are not many results. However, consider
complete warped product metrics on JR.^3 , of the form
(B.4) g = dr^2 + w(r)^2 gcan,
where gcan is the standard metric on S^2. (Recall that the sectional curvatures
v 1 , v2 of such metrics are given in terms of w by (1.58).) For such metrics, we
can prove convergence to a soliton if the curvature is positive and bounded
and the manifold "opens up" like a paraboloid.
THEOREM B.5 (Ivey [220]). Let g be a complete metric on JR.^3 of the
form (B.4). Suppose
(B.5)and suppose(B.6)for positive constants C and Z, and(B.7) liminf (
88
r->oo r w^2 ) > 0.Then the solution of the Ricci fiow with g(O) = g exists for all time. If, in
addition,(B.8) limsup (
88
w^2 ) < oo,
T---700 r
then the fiow converges to a rotationally symmetric steady gradient soliton,
in the sense of the C^00 -Cheeger-Gromov topology (see Theorem 3.10).Intuitively, the condition (B. 7) means that the area of a sphere centered
at the origin grows at least as fast as for a paraboloid, while (B.8) means
that the sphere area grows no faster than a paraboloid. In other words, the
metric becomes fl.at as r-+ oo, but not too fl.at. (By contrast, the result of
Shi [330], which gives convergence of the Ricci fl.ow to a fl.at metric, assumes
that sectional curvatures fall off like r-(2+c) .) The condition v2 ::; Zv1 means
that as the sectional curvature v1 along the planes tangent to the spheres
becomes fl.at as r -+ +oo, the sectional curvature v 2 of the perpendicular
planes also becomes fl.at. Of course, for the Bryant soliton of subsection 3.2
of Chapter 1, v2 falls off much faster than v1. However, it is not difficult to
construct other metrics that satisfy the conditions in the theorem; see [220]
for details.
For metrics on JR.^3 of the form (B.4), the warping function satisfies a
quasilinear heat equation
aw fl 1-(w')^2
8t = w - w = -(v1 + v2)w.