1547671870-The_Ricci_Flow__Chow

272 9. TYPE I SINGULARITIES

PROOF. We may assume that supxEMn IRm(x,t)I is finite fort::; 0.

Because Re 2': 0, we can estimate the growth of the scalar curvature R by

:t R = flR + 2 IRcl

2

::; flR + 2R^2.

Hence Rmax (t) ~ supxEMn R (x, t) ::; C (n) supxEMn IRm (x, t)I satisfies the

differential inequality

So for all t < 0, we have

1

Rmax (t) 2': (Rmax (0))-l - 2t.

Letting t--> -oo, we obtain

which implies the result.

1

lim t-+-00 inf ( It I Rmax ( t)) 2': - 2 ,

D

REMARK 9.18. Because nonnegative Ricci curvature is preserved only

in dimension n = 3, one usually must assume that the curvature operator is

nonnegative in order to apply the lemma. (See Section 2.)

We are now ready to state and prove our main assertion. We shall say

that (x, t) is an ancient Type I c-essential point if

IRm (x, t)l · ltl 2': c > 0.

By modifying the proof of Theorem 9.9, we obtain the following:

THEOREM 9.19. Let (M^3 , g (t)) be a complete ancient solution of the

Ricci flow with positive sectional curvature. Suppose that

sup It!". R (x, t) < oo

M3x(-oo,OJ

for some r > 0. Then either (M^3 , g (t)) is isometric to a spherical space

form, or else there exists a constant c > 0 such that for all T E ( -oo, OJ and

8 > 0, there are x E M^3 and t E (-oo, T) such that (x , t) is an ancient Type

I c-essential point and a 8-necklike point.

PROOF. We will show that if for every c > 0 there are TE (-oo, OJ and

8 > 0 such that there are no ancient Type I c-essential 8-necklike points

before time T, then (M^3 , g(t)) is a shrinking spherical space form.

By hypothesis on the ancient solution, there is r > 0 small enough that

K ~ sup lt l.,, R (x, t) < oo.

M^3 x(- oo,0]