14 1. RICCI SOLITONS
where N = 81 , and it is natural to assume that Rc(g) + \7^2 f = 0 for a radial
function f(r). Using (1.38), (1.39) with p = 0 and n = 1 gives
(1.40) wf" - w" = 0 = w( w' f' - w").
Integrating w f" - w' f^1 = 0 gives
(1.41) J' = 2aw
for some constant a, whereupon w' f' - w" integrates to
(1.42) w' - aw^2 = b.
Using the closure conditions w(O) = 0, w'(O) = 1, we get b = 1. Integrating,
we obtain a smooth odd function w(r) whose type depends on the sign of a:• for a= 0, w(r) = r, giving the flat metric;
1- for a= a^2 , w(r) = - tan(ar);
a
1 - for a= -a^2 , w(r) = -tanh(ar).
a
The third case is the cigar soliton (see [180], [373])
(1.43) 9cig = dr^2 + 2 1 tanh^2 (ar)de^2 ,
a
where by (1.41) we see that the potential function may be taken to be
f ( r) = -2 log cash( ar). The Gauss curvature is K = 2a^2 sech^2 ( ar) > 0.
In the second case, the metric (see [125])(1.44) 9xpd = dr^2 + 2 1 tan 2( ar ) de^2 ,
a
0 < r < n/ (2a), which we call the exploding soliton, is not com-
plete, since tan( ar) ----* oo at a finite distance away from the origin. Thepotential function is f (r) = -2logcos (ar) and the Gauss curvature is
K = -2a^2 sec^2 (ar) < 0.
The above steady gradient solitons are defined on topological disks. By
taking b = -1, we have the following steady solitons on the punctured disk.
1
For a= -a^2 , taking w(r) = - coth(ar) yields
a
1
g = dr^2 + 2 coth^2 (ar)de^2 ,
awhich has potential f (r) = -2logsinh(ar) and K = -2a^2 csch^2 (ar) < 0.
1
For a= a^2 , taking w(r) = -cot(ar) yields
a
1
g = dr^2 + 2 cot^2 (ar)de^2 ,
a
which has potential f (r) = 2 log sin( ar) and curvature K = 2a^2 cot( ar) >
- Both of the above metrics are incomplete because of their behavior near
r = 0.