1547671870-The_Ricci_Flow__Chow

212 6. THREE-MANIFOLDS OF POSITIVE RICCI CURVATURE

Let x, y E M^3 and T/ > 0 be given. Then by (6.53), (6.55), and (6.56), there

exists TT/ E [O, T) such that for all times TT/ :S t < Tone has

v (x, t) :::=: (1 - TJ) >. (x, t)

1 - T/

::::: -

3

-R(x,t)

:::::(l~TJ)2R(y,t)

::::: (l~TJ)

2

[.X(y,t)+2(1-TJ).X(y,t)]

::::: (1 - TJ)^3 >. (y, t).

The claim follows by taking the infimum over all x E M^3 and the supremum
over ally E M^3. 0

This result lets us conclude in particular that g (t) is approaching an

Einstein metric uniformly as t / T.

COROLLARY 6.56. If (M^3 , g (t)) is a solution of the unnormalized Ricci

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

. ( 1:Rc1

2

hm sup -R 2 ) = 0.

t/T xEM3

PROOF. Applying the local pinching estimate of Theorem 6.30 in the

form (6.37) shows that there are positive constants C and J such that

0

IRcl2 < CR-5 < R-5.

R2 - - mm

Since limt/T Rmax (t) by Theorem 6.54 and limvr (Rmin/ Rmax) 1 by

Lemma 6.55, the result follows. 0

9. Properties of the normalized Ricci flow

We now have at our disposal all the facts about the unnormalized Ricci
fl.ow that we will need to prove Theorem 6.3. But since the theorem is stated
for the normalized fl.ow, we must convert to that flow in order to complete
its proof. In this section, we shall study how this is done and will then show
that the normalized flow exists for all time and asymptotically approaches
.an Einstein metric. In the final part of this chapter (Section 10, below) we
will prove that the convergence is in fact exponential.

9.1. Equivalence of the flows under rescaling. We begin by show-

ing that the unnormalized and normalized Ricci flows differ only by a rescal-

ing of space and time. Specifically, suppose that (Mn,g (t)) is a solution of

the unnormalized Ricci fl.ow

a

-g = -2Rc

at