214 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE
This shows that ~ = -b ~t, hence by (6.57) that g evolves by the normalized
Ricci fl.ow
f) - - 2p -
f)f g = -2Rc + -;;;: g.
9.2. Long-time existence of the normalized flow. Let (M^3 , g (t))
be the unique solution of the unnormalized Ricci fl.ow starting on a 3-
manifold of positive Ricci curvature. By Theorem 6.54, g (t) becomes sin-
gular in the sense that it exists on a maximal time interval 0 :::::; t < T < oo
and satisfies limvr Rmax (t) = oo, where Rmax (t) ~ supxEM3 R (x, t). Let
(M^3 , g (l)) be the corresponding normalized solution constructed in Sub-
section 9.1. Then g (l) exists on a maximal time interval 0 :::::; f < T :::::; oo.
We shall now prove that g enjoys long-time existence, namely that T = oo.
THEOREM 6.58. If (M^3 ,g (t)) is a solution of the unnormalized Ricci
flow starting on a compact manifold of positive Ricci curvature, then the
corresponding normalized solution (M^3 , g (l)) exists for all positive time.
and
We begin with some preliminary results. Define
Rmin (l) ~ inf R (x , l)
xEM^3Rmax (l) ~ sup R (x , l).
xEM^3LEMMA 6.59. If (M^3 , g (t)) is a solution of the Ricci flow with initially
positive Ricci curvature, then
foT Rmax (t) dt = 00.PROOF. Since Rmax is a continuous function of t E [O, T), there is a
unique solution p to the initial value problem
{
dp/ dt = 2Rmax · P.
p(O) = Rmax (0)
Since g (t) has positive Ricci curvature for as long as it exists, we have
1Rcj^2 :S R^2 :S R · Rmax, which implies that
:t (R- P) = ~ (R-p) + 2 (1Rcj
2- Rmax · p)
:S ~ ( R - p) + 2Rmax ( R - P) ·
So by the maximum principle, we have R :::::; p for all points x E M^3 and
times 0 :S t < T. This implies in particular that p 2: Rmax, hence by
Theorem 6.54 that lim 7 /T p ( T) = oo. Since for 0 :::::; T < T, we have
p(T) (7d r
logp(O) =Jo dtlogpdt=2}
0Rmax(t)dt,the integral on the right-hand side must diverge. D