- IMMORTAL SOLUTIONS 35
Consider the one-parameter family of expanding metrics g ( t) defined for
t > 0 in the polar coordinate system (r, e) on IR.^2 \ { any ray } by
(2.28) g (t) = t (J(r)^2 dr^2 + r^2 dB^2 ),
where f is a positive function to be determined. We will study the metrics
g ( t) in local coordinates ( z^1 = r , z^2 = e) , instead of using moving frames
as we did for the cigar soliton in Lemma 2.7 of Section 2. Again recalling
the formula
k lke(o 8 8)
rij = 29 [)xi9jl! + oxJ9il! - axe9ij '
we compute that the Christoffel symbols of g are
n1 =!'If rb = o n2 = - r/!2
rr1 = 0
Then the standard formula
e 8e 8e e e
Rijk = [)xi rjk - oxJ rik + rimr_;k - rJmr~shows that the only nonzero components of the Ricci tensor Rjk = RLk of
g are(2.29)2 f' (r)
Ru= R2u = rf (r) and
Rn= R122^1 = --rf'(r) 3.f (r)
Because Re= Kg, where K is the Gauss curvature of g, this shows that
(2.30)
_ 1 J'(r)K(t) = K[g(t)] = -·.
t rf(r)^3
Now let X be the time-dependent vector field defined on IR.^2 \ {O} by
r [)
(2.31) x (t) ~ tf (r). or.Notice that the 1-form ~ metrically dual to X is time-independent:
(2.32) ~ = rf (r) dr.
The components of the Lie derivative £xg of g with respect to X are(2.33a) (£x9h1 = \716 = or [) (r f (r)) - r^1 u 6 = f ( r)(2.33b) (£xg)i 2 = \716 = O
(2.33c) (£xg) 21 = \726 = 0
r2
(2.33d) (£xg) 22 = \726 = -r~26 = f (r) ·We want to determine f so that g (t) evolves by the modified Ricci fl.ow
[)
( 2. 34) ot g ( t) = -2 Re [g ( t) ] + £ x ( t lg ( t).