36 2. SPECIAL AND LIMIT SOLUTIONS
(Compare with the Ricci- DeTurck flow analyzed in Section 3 of Chapter
3.) If we can construct a solution g (t) of the modified Ricci flow (2.34),
we will obtain an expanding self-similar solution g (t) of the Ricci flow as
follows. Observe that there exists a one-parameter family of diffeomorphisms
1Pt : IR^2 ____, IR^2 defined for all t > 0 by
{)
at <pt(p) = -x ( <pt(p) , t).
(Because IR^2 is not compact, long-time existence of 1Pt must be checked
explicitly, in contrast with Lemma 3.15.) Define an expanding self-similar
family of metrics g ( t) by
(2.35) g (t) ~ <p; (g (t)) = t. <p; (! (r)^2 dr^2 + r^2 de^2 ).
Using the identity~I (1Pt^10 1Pt+s) = (1Pt
1)* (~I 1Pt+s) = (1Pt
1
LX(t),
us s=O us s=O
we compute that
:tg (t) = :t (<p;g (t)) = :s ls=O (1P;+s9 (t + s))
= <p; (%tg(t)) + :sls=O (1P;+sg(t))
= <p; { -2 Re [g (t)] + Lx(t)9 (t)} + :s ls=O [ ( 1Pt^1 o 1Pt+s)* <p;g (t) J
= -2Rc [g (t)] + <p; (£x(t)9 (t)) - £[('P!1).x(t)] (<p;g (t))
= -2Rc [g (t)].
Hence the self-similar metrics g (t) defined by (2.35) solve the Ricci flow.
To construct a solution g (t) of the modified Ricci flow, we observe by re-
calling (2.29) and (2.33) that g (t) will solve (2.34) if and only if the following
system of equations is satisfied:
2 8 J'(r)
f(r) = otg11 = -2R11 + 2Y'16 = -2 r f (r) + f (r)2 8 r f' ( r) r^2
r = ~922 = -2R22 + 2Y'26 = -2-.-- 3 + -1
( ) ·
ut f (r) r
Remarkably, multiplying the second equation by f^2 /r^2 gives the first, so
that this system is equivalent to the single first-order ODE
(2.36)Integrating this separable equation using partial fractions gives the relation
r^2 1 f (r)
(2.37) 4 = C - f (r) +log If (r) - 11 ·