1547671870-The_Ricci_Flow__Chow

3. POSITIVE SECTIONAL CURVATURE DOMINATES 261

COROLLARY 9.7. In dimension n = 3, every Type I limit of a Type I sin-

gularity or Type II limit of a Type Ila singularity has nonnegative sectional

curvature for as long as it exists.

PROOF. Let (M^3 ,g (t)) be a singular solution with blow-up time T <

oo. As in Section 3.1 of Chapter 8, let (xi, ti) be a sequence of points and
times chosen such that ti / T and

JRm (xi, ti)I 2: c2 sup JRm (x, ti)I 2: c2c1 sup JRm (x, t)I,

xEM^3 (x,t)EM^3 x[si,ti]

where (ti - si) supxEM3 IRm (x, ti)I --; oo. Suppose that a subsequence of

the pointed solutions (M^3 , gi (t), xi) defined by

gi(t) ~ IRm (xi,ti)I · g (ti+ JRm (:i, ti)I)

converges to a complete pointed solution (M~, g 00 (t), x 00 ) of the Ricci flow.

Then by Proposition 8.20 and (8.18), g 00 (t) is an ancient solution satisfying

sup I Rm (x, t) I < oo

xEM~^00

for all t ::::; 0. By Corollary 9.8, the sectional curvatures of g 00 are nonnega-

tive. D

Another consequence of the pinching estimate in Theorem 9.4 is that

ancient 3-dimensional solutions of the Ricci flow have nonnegative sectional

curvature.

COROLLARY 9.8. Let (M^3 , g (t)) be a complete ancient solution of the

Ricci flow. Assume that there exists a continuous function K ( t) such that

Jsect [g (t)]I ::::; K (t). Then g (t) has nonnegative sectional curvature for as

long as it exists.

PROOF. If (M^3 , g ( t)) is a solution of the Ricci flow that exists at least

for 0 ::::; t < T with

0 > vo ~ inf v ( x, 0) ,

xEM^3

then the family of metrics g ( t) defined on M^3 by

9 ( t) ~ I vo I · g ( I : 0 I )

is a solution of the Ricci flow with v 2: -1 at t = 0. So at any (x, lvol t) E

M^3 x [O, T), Theorem 9.4 gives the estimate

R (x, lvol t) 2: Iv (x, lvol t)I ·[log Iv (x, lvol t)I +log (1 + lvol t) - 3]

wherever v (x, Jvol t) < 0. (Maximum principles work on noncompact as

well as on compact manifolds. We will provide a complete proof of this fact