1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. PERELMAN'S K:-SOLUTION ON THE n-SPHERE 101

where Re - denotes the Ricci tensor of ?fL· We then compute

-d d Rmax ,,.._ :S 2 max 1-12 Re

t xEO'

= 2max (IRc-.!.ii9L1

2

+ .!.ii

2

xEO' n n )

::::; ~ (1+11) ( Rmax )2,

where n' = n' (t) ~ { x E sn: R (x, t) = Rmax (t) }· Integrating the inequal-

ity

d ( ,,.._ )-1 ( ,,.._ )-2 d ,,.._ 2

dt Rmax = - Rmax dt Rmax 2: --;;;, (1+77)

from -1 to 0, while using limt-+O Rmax (t) = oo, we obtain

(19.36)

"' n

Rmin (-1) 2: 2 (l + ry)"

By applying (19.28) at t = -1, i.e.,

maxsn sect (g L ( -1))

8

minsn sect (?JL(-1)) ::::;

1

+

to (19.35) and (19.36), i.e.,

n ,,.._ n

2 (l+ry) :SRmin(-1) :S 2'

we conclude that the sectional curvatures of ?fL (-1) are close to 2 (n~l);

intuitively, ?f L ( -1) is metrically close to the round sphere metric of radius

J2 (n -1). ·

(ii) Second we note the changing distances estimate for ?fL(t):

f) ,,.._

(19.37) at dt(X, y) 2: -4 (n - 1)

-1-to(L) n

(L)

for t ::::; -1 and x, y E S ,

t - to

where dt denotes the distance function for ?fL (t).
To see this, recall that by Theorem 18.7(2), for a solution (Mn, g (t)) of
the Ricci flow, if

Rc(x, to) ::::; (n - l)K

for all x E Bg(to) (xo,K-^112 ) UBg(to) (x1,K-^1 l^2 ), then the distance function
dg(t)(xo, x1) satisfies the following differential inequality (at time t =to):

:t lt=to dg.(t) (xo, x1) 2: -4 (n - 1) VK.

To obtain (19.37), by Rm 2: 0 and (19.33), it suffices to choose

K ( t) = -1 - to ( L).

t - to (L)