26 1. RICCI SOLITONSis positive everywhere else. The curvatures approach zero as r -----t +oo, so R
has a positive maximum value at some point. Because f' = (y - nx)/w ispositive for r > 0, this must occur at the origin. D
We end this section by noting that the preceding construction can be
generalized to the case where the fiber is a product of an Einstein manifold
and a sphere (see [219]).THEOREM 1.38 (Steady solitons on doubly-warped products). Given an
Einstein manifold (Nm, 9N) with positive Ricci curvature, there exists a 1-parameter family of complete steady gradient soliton metrics on JRn+l x N
in the form of doubly-warped products^11 :g = dr^2 + v(r)^2 9can + w(r)^2 gN,
where 9can is the standard metric on sn' n 2: 1.
These solitons have positive Ricci curvature when r > 0. ·However, the
sectional curvature along the copies of N can be negative.5. Rotationally symmetric expanding solitons
In Section 4 in Chapter 2 of Volume One, we described the construc-
tion of a 1-parameter family of rotationally symmetric expanding gradient
Ricci solitons on JR^2 (see also Gutperle, Headrick, Minwalla, and Schome-
rus [174]). These solitons are asymptotic to a cone (with any cone angle
in the interval (0, 27r) possible) and have curvature exponentially decaying
as a function of the distance to the origin (see Exercise 4.15 and Corollary
9.60 of [111]). In this section we consider the problem of constructing rota-
tionally symmetric expanding gradient Ricci solitons on JRn+l for n 2: 2. In
particular, we generalize the construction from the previous section on the
Bryant soliton to show that there exists a 1-parameter family of rotationally
symmetric complete expanding gradient solitons on JRn+l.5.1. Setting up the ODE for expanding gradient solitons. To
start, we begin with the expanding gradient soliton condition
c
Rc(g) + \7\7 f + .\g = 0, .\ = -> 0.
2
Substituting (1.38) and (1.39) into this equation gives
w"
0 = -n-dr^2 + (p -ww" - (n - l)(w')^2 ) g
w
+ (f" dr^2 + ww' f '9) + .\ ( dr^2 + w^2 g) ,
where w(r) is the warping function and g is the metric on sn with Einsteinconstant p. Taking p = n - 1 and collecting the components of the metric
(^11) This construction was inspired by a similar construction for Einstein metrics by
Berard-Bergery [24].