1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

72 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

We can estimate f (q, 1) by the reduced length of the concatenated path

'Y1 U 'Y2 : [O, 1] ---+ M:

1

f ( q, 1) :S ~ lo

2

Vs ( R ( "(1 ( s) , s) + I 'Yi ( s) I;( 8 )) ds

• ~ 1

1

Vs ( R b2 (s), s) + l'Y; (s)l;(s)) ds

2

1 fl

:S c4 (n, A)+ 2 h Vs ( n (n - 1) + 16e^2 (n-l)) ds

2

~ c5 (n,A).

We have proved

f (q, 1) :::; c5 (n, A) for any q E Bg(l)(xo, 1).

Now we can estimate the reduced volume V (1) from below:

v (1) ;:::: r (41fT)-~ e-£(q,T)dμg(T) (q)

JB 9 (1)(xo,l) T=l

2: (47r)-~ e-c5(n,A) { dμg(l)

J Bg(l) (xo,1)

;:=:: (47r)-~ e-c^5 (n,A)A-1.

Now (18.63) is proved by taking c3 (n, A) ~ (47r)-~ e-c^5 (n,A) A-1; hence

Theorem 18.36 is proved.

5.3. Proof of Lemma 18.38.

(i) This standard estimate holds since (18.65) implies

I :tg (t) (W, W) I = 2 IRcg(t) (W, W) I :::; 2 (n - 1) JWJ~(t)

for any t E [O, 1] and any tangent vector Won Bg(O)(xo, 1) (see also (3.3) in
Part I ).
(ii) Let 'Y (s), 0 :S s :S so, be any unit speed geodesic, with respect to
g (t), such that"( (0) = xo and so < e^1 -n. It follows from (i) that

Lg(O) ("!) :S en-l so < 1.

In particular Bg(t)(xo, e^1 -n) C Bg(o)(xo, 1).

(iii) We first define a smooth nondecreasing function

q) : IR ---+ [1, oo]