1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. ESTIMATES FOR CHANGING DISTANCES 47

Combining this with (18.12), we obtain

(! -~g(t)) dg(t)(x,xo)/t=to 2: -(n-1) (~Kro + : 0 ).

This completes the proof of part (1).

(2)(a) Let 'Y: [O, so] ----+ M joining xo to x be as in part (1), where now

we write x = x1, and let s1 E [ro, so - ro] be such that

'Y(s1) ¢:. Bg(to) (xo, ro) U Bg(to) (x1, ro)
(here we used dg(to) (xo, x1) 2: 2ro). Then the function h : M ----+ [so, oo)
defined by

h(x) = dg(to) (xo, x) + dg(to) (x1, x)

attains its minimum so at 'Y(s1) and hence

(18.13)

in the barrier sense.

Arguing as in the proof of part (1), if we define (: [O, s 1 ] ---t [O, 1] by

( ( s) = ro I _ s _ ro,

{

s 'f 0 < <

1 if ro < s S s1,

then we obtain (in the barrier sense)

~g(to)dg(to) ( xo, 'Y( s1))

r

1

S - lo Rc('Y'(s),"f'(s))ds + (n -1) (2 1)

3 Kro + ro.

On the other hand, if we define ( : [s1, so] ----+ [O, 1] by

{

1 if s1 S s S so - ro,

( ( s) = so-s ro if s 0 - r 0 < s < - s o,

then we obtain (in the barrier sense)

~g(to) dg(to) (x1, 'Y( s1))

l

S - so (2 1)

81

Rc('Y'(s), "(^1 (s))ds + (n -1)

3

Kro + ro.

From (18.12), we have

0 S ~g(to)dg(to)(xo, 'Y(s1)) + ~g(to)dg(to)(x1, ')'(s1))

(18.14) S - lo (8° Rc('Y'(s),')'^1 (s))ds + 2(n-1) (2 1)

3 Kro + ro

s aa I dg(t)(xo,x1) + 2(n -1) (-3

2

Kro +I_).

t t=to ro

Part (2)(a) follows.