1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. ALMOST K;-SOLUTIONS 133

Note that^9

(20.16)

Bgk(t*k) (Y*k' 1

1

0 Ak

12

R;

1

l

2

(Yk' tk)) c Bgk(t*k) ( xok, ~~~).

Since Re 2:: 0, (20.16), and dgk(t*k) (Y*k' Xok) < !, we have by the Bishop-

Gromov volume comparison theorem that^10

Volgk(t.k) Bgk(t*k) (Y*b fo-Ak/2 R;1/2(Y*k' t*k))

(

lOAk^1 1/2 Rgk -1/2( Y*k' t*k) )n

Volgk(tk) Bgk(tk) (Y*k' !)

• (!t

Volgk(t.k) Bgk(t*k) (xok, i). (-

3

1)n

2:: (i) n

2:: 3-nvolgk(t*k) Bgk(t.k) (xok, 1)

2:: 3-nwo,

where we used (20.14) with t = t*k to obtain the last inequality.

On the other hand, we can apply Proposition 20.4 to gk (t) ~ gk (t + t*k),

the geodesic ball Bgk(t*k) (Y*k' loAk^12 R;k

1

!

2

(Y*k' t*k)), and the time interval

[-DkR;k^112 (Y*k' t*k), OJ, to obtain that for every c > 0 there exists A ( c) < oo

(^10) Since B 9 k(t.k) (xok, !~6) C B 9 k(t.k) (xok, 1) C B9k(o) (xok, 1), which is compactly
contained in Mk, this enables us to apply the volume comparison theorem.