1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. PERELMAN'S K-SOLUTION ON THE n-SPHERE 99

On the other hand, since the initial scalar curvature satisfies

R(gL (0)) :2: c > 0,

independent of L,^21 by applying the weak maximum principle to the evolu-

tion inequality ftR :2: !J.R + ~ R^2 for the scalar curvature, we obtain

This implies

n

R (9L (t)) :2: '.!! _

2

(

c

n

TL S 2c·

To summarize, 9L (t) forms a singularity at a time TL E [c (n), ~] and

there exists K,o > 0 such that 9L (0) is K,o-noncollapsed for all L > 1.

By Perelman's weakened no local collapsing theorem (see §7.3 in Perel-

man [152] or Theorem 8.26 in Part I, which is restated as Theorem 19.52

below), we conclude that there exists K, > 0 independent of L such that 9L(t)

is K,-noncollapsed at any scale 0 < r :::::; yfij-for all L > 1. Note that since
we have Rm :2: 0, weakly K,-noncollapsing (seep. 401 of Part I) is eqμivalent
to K,-noncollapsing; moreover, from the proof of Theorem 8.26 in Part I, it
is clear that the dependence of K, on sup Mx [o,~] Rc9(t) can be replaced by

supM [o 1 ] Rc 9 L(t),^22 which is bounded independent of L here. Since

X '16C(n)
the property of K,-noncollapsing is invariant under scaling, the solution 9L(t)

is K,-noncollapsed at any scale 0 < r S .J 4 (T~~tL) for all L > 1. D

REMARK 19.41. The reason for using the weakened no local collapsing
theorem (proved using the reduced volume monotonicity) instead of the
no local collapsing theorem (proved using entropy monotonicity) is that by
using the former result one can show that K, is independent of L.

STEP 5. The rescaled solutions tend to ancient as L--+ oo.

Claim 1. In definition (19.30) we have

(19.31)

that is,

(19.32)

lim to (L) = -oo,

L-+oo

(^21) Note that for n = 2 this positive lower bound for R does not hold for the above
construction since a 2-dimensional cylinder is flat.
(^22) To wit, the factor ~ in t can be replaced by any number in the interval (0, 1), in
particular 16 J(n).