- IMMORTAL SOLUTIONS 37
Notice that f (r) ----? 1 is possible only if r ----? oo, so that we either have
0 < f (r) < 1 or 1 < f (r) for all r. But combining equations (2.30) and
(2.36) shows that the Gauss curvature of g (t) is
(2.38)
1 1-f
K(t)=-·-.2t f
So in order to have a metric of positive curvature, we must find a solution
satisfying 0 < f (r) < 1. Such a solution does in fact exist, and can be
given explicitly. Recall that the function w : (0, oo) ----? (0, oo) defined by
w ( x) = xex is invertible; its inverse is the Lambert-W (product log) function
W: (0, oo)----? (0, oo). One can verify directly that
1
(2.39)
f(r)= 1 + W (((^1 - 1 ) exp ((1 ( - 1 ) - r2))
4is the solution of (2.37) satisfying f (0) = ( E (0, 1).
Following [55], we shall now demonstrate how one might arrive at therepresentation (2.39) for funder the Ansatz that 0 < f (r) < 1. Define
1
F (r) ~ f (r) - 1,
noting that the Ansatz forces F (r) > 0 for all r ::'.'.'. 0. Then equation (2.37)
is equivalent to
r2
4= C' - F ( r) - log F ( r) ,where C' = C + 1, which we write asC' - r: = F(r) +logF(r) =log (F(r)eF(r)).
Observing that C' = log (F (0) eF(Ol), we can put this into the form
(2.40) F(r)eF(r) = eC'-r
2
/^4 = F(O)exp (F(O)- r:).By the Ansatz, we ha~ F (r) > 0 for all r ::'.'.'. 0. Thus we may apply W to
both sides of equation (2.40), obtainingF ( r) = W ( F ( 0) exp ( F ( 0) - r: ) ) ,.whence equation (2.39) follows immediately.Note that limr->oo F (r) = 0, so that limr_, 00 f (r) = 1. Note too that
equation (2.38) implies that limr_, 00 K (t) = 0 for all t > 0. This reflects
the fact that the metrics g (t) are asymptotic at infinity to a flat cone.