1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

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(18.9)

(18.10)

18. GEOMETRIC TOOLS AND POINT PICKING METHODS

(a) If dg(to) (xo, x1) 2: 2ro, then the time derivative of the distance

function has the lower bound

a

a I dg(t)(xo,x1) 2: -2(n-1) (~Kro +-2:_).

t t=to 3 ro

(b) If dg(to) (xo, x1) < 2ro, then

: I dg(t)(xo,x1) 2:-2(n-l)Kr 0.

t t=to

Clearly, in either case we have

(18.11) aa I dg(t)(xo, x1) 2: -2(n - 1) (Kro + _!_).

t t=to ro

PROOF. (1) We first observe that as a consequence of Corollary 18.2,

Lemma 18.4, and the fact that Z (to) is compact, there exists I E Z (to)

such that

(18.12) % I dg(t)(x,xo) 2:-lRcg(to) (1'(s),1'(s))ds.

t t=to 'Y

Since

so ~ dg(to) (x, xo) > ro

by assumption, we may choose

'

( ( s) = J :a if 0 ::; s ::; r^0 ,

l 1 if ro < s ::; so

in (18.4), so that

1

ro (n-1 s2 )

~g(to)dg(to)(xo,x) :S - 2 - - 2Rcg(to)b'(s),1'(s)) ds

o ro ro

We simplify this as

~g(to)dg(to) (xo, x)

1

80

- Rcg(to) (!' (s ), 1' (s) )ds.

ro

• ::; - 1

80

Rcg(to) (!' ( s), 1' ( s) )ds

• fora ( Rcg(to) ( 1' ( s), 1' ( s)) ( 1 - :; ) + n ~

1

) ds

:S - Rcg(to)(1'(s),1'(s))ds + (n -1) -Kro + -

1

80 (2 1)

o 3 ro

since Rcg(to) ::; (n - l)K along 1lro,ro] C Bg(to) (xo, ro).

Therefore, in the barrier sense, we have

~g(to)dg(to)(xo,x) :S - Jo (8° Rcg(to)(1'(s),1'(s))ds + (n -1) (2 1)

3

Kro + ro.