1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

134 20. COMPACTNESS OF THE SPACE OF /;;-SOLUTIONS

such that for sufficiently large k

• 1/2

Volgk(t.k)Bgk(t.k)(Yk,A(c)R~ (Yk't*k)) <

(A (c) R;;,^112 (Y*k' t*k)) n - c.

If k is large enough, then lo Ak/^2 > A ( c) and, by the Bishop-Gromov volume

comparison theorem, we have^11

u vo (^1) 9k(t.k) 9k(t.k) B ( Yk' 10 1 Al/2R-1/2( k 9k Yk' t *k ))

3-nw < ~'--~~----'-

O - ( 1 1/2 -1/2 )n

10 Ak R^9 k (Y*k' t*k)

Vol 9 k(t.k) B 9 k(t.k) ( Y*b A -(c) R~ 1/2 (Y*k' t*k) )

s (A(c)R~^112 (Yk'tk))n Sc.

We obtain a contradiction if we choose say c =^3 -;wo. Proposition 20.6(a)

is proved.

2.2.3. Proof of Proposition 20.6(b} with ro = 1.

Let (Mn, g ( t)) be a solution of the Ricci flow satisfying the assumptions

of Proposition 20.6(b) with ro = 1. Let ro (w) > 0 be a constant depending

only on w which will be specified later (see (20.20) below). Suppose that^12

-to ~ ro (w) and define ti E [to, O] to be the smallest constant such that^13

19

dg(t) (y, xo) S 80

for ally E Bg(O) (xo, ~) and t E [t1, OJ.

STEP 1. We shall prove the following.

Claim 1. We have

(20.17)

fort E [t1, O].

It follows from the nonnegativity of the Ricci curvature and the Bishop-

Gromov volume comparison theorem that fort E [t 1 , OJ,

1

Volg(t) Bg(t)(xo, 1) ~ Vol 9 (t) B 9 (o)(xo, 5)

1

~ Vol 9 (o) B 9 (o)(x 0 ,

5

)

~ ( ~) n Volg(O) Bg(O) (xo, 1).

From the assumption (20.12), with ro = 1, we obtain (20.17).

(^11) Here again, we use B
9 k(t.k) (Y•k, f"oA~^12 K;k^112 (y.k,t•k)) C B 9 k(o) (xok,l) is com-
pactly contained in Mk, as we remarked above.
(^12) 0therwise there is nothing to prove.
(^13) Clearly there exists such a t 1.