1547671870-The_Ricci_Flow__Chow

216 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

R = R/1/J, and dp, = 1jJnl^2 dμ = 1jJ^312 dμ. Thus for any T E [O, T), we have

j

·T JTf Rd

p (t) dt = M^3 μ dt

o fMsdμ

0

f

f J nl•RnJ,-3/2 d -

= M3 'f/ 'f/ μ 1/J-1 dt

f Ms 1/J-3/2 dP,

0

= lf p(t) dt,

where f = J; 1/J (t) dt. Now since

Rmin (t) :SP (t) :S Rmax (t),

we apply Lemmas 6.55 and 6.59 to conclude that the left-hand side diverges
as T / T. Hence the right-hand side must also diverge as f / f'. But by

Lemma 6.60, there exists C > 0 such that

lf p (t) d[ :S lf Rmax (t) dt :S Cf.

Hence f' = oo. D

Noting that the expression in Corollary 6.56 is invariant under dilation

of the metric, we obtain the following convergence result for free.

COROLLARY 6.61. If (M^3 , g (t)) is a solution of the unnormalized Ricci

flow on a compact manifold whose Ricci curvature is positive initially, then

the corresponding normalized solution g (t) exists for all positive time and

asymptotically approaches an Einstein metric uniformly in the sense that

Jim (sup l~ct) = 0.

t->oo xEM3 R

In Section 10, we shall show that the convergence is exponential.

9.3. Evolution of curvature under the normalized fl.ow. To aug-

ment the scaling arguments employed above, it will be useful in the sequel

to observe directly how various geometric quantities evolve under the nor-

malized Ricci fl.ow. To wit, suppose that we are given a solution of the

normalized Ricci fl.ow

a

atg = -2 Re +rg

on a manifold Mn, where r is the function of time alone defined by

(6.61) r ( ) t =. 2 ( ) - p t _ - ----'--="^2 j Mn ~-R dμ -

. n - n fMn dμ