1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

102 19. GEOMETRIC PROPERTIES OF i;;-SOLUTIONS

(iii) The diameter of 9L (to (L)) does not exceed -8 (n -1) to (L).

By integrating (19.37) from to (L) to -1, we obtain

(L1(x,y)-~o(L)(x,y) = {-l ~ dt(x,y)dt

lto(L) ut

> {-l -4 ( n - 1) - l - to ( L) dt

• lto(L) t - to (L)

= -8 (n -1) (-1-to (L)),

so that

(19.38) ~o(L)(x,y):::::; ;L1(x,y) + 8 (n -1) (-1-to (L)).

On the other hand, since the sectional curvatures of 9L (-1) are close to that

of the round sphere of radius .J2 (n - 1) (see the end of (i) in the proof),

by Myers's theorem, we have

cL1(x, y) :::::; 2.JrL=17r

for any x, y E sn so long as c is sufficiently small. Now (iii) follows from

(19.38):

~o(L)(x, y) :::::; 2.JrL=l7r - 8 (n - 1) - 8 (n - 1) to (L)

:::;-8(n-l)to(L).

We now finish the proof of Claim 1. Since diam (9L (0)) 2 L, by our

construction of 9L(t) we know that the initial diameter has the lower bound

diam(gL(to (L))) 2 L

vTL-tL

Hence

and we have

1

TL2 8 (n-l)LVTL-tL.

Recalling that TL E [c (n), ~],we obtain the key estimate

lim (TL - tL) = 0,

L--+oo

that is,

TL

to (L) = -T --+ -oo as L--+ oo.

L -tL

This finishes the proof of Claim 1.