1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

98 19. GEOMETRIC PROPERTIES OF A;-SOLUTIONS

By Theorem 6.3 in Volume One and Theorem 11.2 in Part II (i.e.,
Hamilton's convergence theorem for solutions of the Ricci fl.ow with pos-
itive Ricci curvature on closed 3-manifolds [88], his 4-dimensional result
[89], and Bohm and Wilking's corresponding result in all dimensions for so-
lutions with 2-positive curvature operator [16], respectively), we know that
under the Ricci fl.ow, the metric 9L (t) shrinks to a round point as t---+ TL.

For any given 6 > 0 sufficiently small, let tL E (0, TL) be a time such that

the global sectional curvature ratio satisfies

(19.28)

maxsn sect (gL(t))

A(gL(t))~ mmsn. sect ( 9L ()) t ::;1+0 fortE[tL,TL)

with equality (i.e., = 1+6) holding at the time tL. (Note that A (g) - 1 is

a scale-invariant measure of how close the metric g is to constant sectional

curvature.) The existence of such a time tL E (0, TL) follows from the fact

that the solution 9L (t) of the volume normalized Ricci fl.ow with 9L (0) =

9L (0) converges to a round sn. Note that for any c > 0 we have

lim maxsn sect (9£ ( t)) = 00

t-+0 minsn sect (9£ ( t))

(the principal sectional curvatures, equal to ~ of the eigenvalues of the Rie-

mann curvature operator, at t = 0 are equal to 2 (n1:_ 2 ) or 0 on the cylinder

part).

We consider the family of rescaled and time translated solutions

(19.29) gL(t) ~ T l 9L(TL +(TL - tL) t)
L-tL
for TL+ (TL-tL)t E [O,TL), where LE (1,oo). The initial time for the
solution 'fh(t) is

(19.30)

whereas the singularity time is 0.

STEP 4. The normalized family is uniformly K,-noncollapsed.

LEMMA 19.40. There exists "" > 0 independent of L such that 9L(t) is

K,-noncollapsed at any scale 0 < r :::; J 4 (T~~tL) for all L E (1, oo).

PROOF. Since by (19.27), for the original solution 9L (t), the curvature

tensor satisfies IRml (gL (0)) :::; C (n) for some C (n) < oo independent of L,

we know that

TL 2:: c(n)

for some constant c ( n) > 0 by Hamilton's doubling time estimate for the

norm of the curvature tensor (see Corollary 7.5 in Volume One or the slightly
more precise Lemma 6.1 in [45]).^20

(^20) In particular, IRml (gL (t)) ::::; 20 (n) fort E [O, 1/ (160 (n))].