18 1. RICCI SOLITONS4.1. Setting up the ODE for Bryant solitons. With p = n - 1,
substituting (1.38) and (1.39) into the steady gradient soliton condition
Rc(g) + V'V' f = 0 gives the following system of ODE for w and f:
fl
(1.45) f" = n'!!!_, ww' f' = ww" - (n - 1)(1 - ( w')2).
wSince R = -.6..f for a steady gradient soliton, Proposition 1.15 implies that
.6..f - IV' fl^2 = C, a constant. On the other hand, tracing (1.39) gives
I!'
.6.. f = f" + n '!!!_____ •
w
Using this expression for .6..f, we obtainw^2 !" + nww' f' - w^2 (!')
2= Cw^2.
Eliminating f" and w" using (1.45) gives a first integral of our ODE system:
(1.46) 2nww' f' + n( n - 1) (1 - ( w')^2 ) - w^2 (!')^2 = Cw^2.
The analysis of solutions to (1.45) is simplified by using variables that are
invariant under the symmetries of the system (i.e., translating r, translatingf, and simultaneously scaling r and w). We choose new dependent variables
x and y and independent variable t (not to be confused with time), such
that
x=:=w, • I y ~ nw' -wf', dt~ -. dr
w
From (1.46) we have
(1.47) nx^2 - y^2 + n (n - 1) = Cw^2.
Then by (1.45)
dx
dt = ww" = ww' f' + (n - 1)(1-(w')^2 ),
dydt = wy' = -ww' f'.
Thus the ODE system (1.45) becomes(1.48)dx 2
-=x -xy+n-1
dt 'dydt = x(y - nx).
Substituting C = 0 in the first integral (1.47) gives an invariant hyperbola
y^2 - nx^2 = n(n - 1) for the system (1.48); see Figure 1 on the next page.
REMARK 1.30. When n = 1, equation (1.48) becomes
dx dy(1.49) dt =x(x-y), dt =x(y-x),
which implies x + y = canst. Assuming, x, y , 1 as t , -oo, we have
dx
dt = 2x (x - 1). The solution of (1.49) with x (t) decreasing is given by
1 1
x = 1 + e2t ' y = 2 - 1 + e2t.