1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. THE COMPACTNESS OF />;-SOLUTIONS 147

By Lemma 18.15 there exists Wk E B 9 k(o) (xk, r*) and Sk E (0, r* + 1)

such that

(a) R 9 k (wk, 0) s~ -+ oo as k-+ oo and

(b) R 9 k (x,O)::::; 2R 9 k (wk,O) ~ 2Qk for all x E B 9 k(o) (wk,sk)·

From the proof of Lemma 18.15 it is clear that

R 9 k (wk, 0) 2: (r* + 1)-^2 sup R 9 k (x, 0).

xEB 9 k(o) (xk,r*)

Hence akRgk (wk, 0)-+ oo.

Now we can apply Proposition 20.4 on the parabolic cylinder

B 9 k(o) (wk, sk) x (-ak, OJ

to obtain that for any c: > 0 there exists A (c:) E (0, oo) such that

Vol 9 k(O) B 9 k(o) (wk, A (c:) Qk,

1

/

2

)

( )

n < c

A (c:) Qk,1;2 -

for k large enough.

On the other hand, for any Wk E B 9 k(o) (xk, r*) we have

B 9 k(o) (xk, ro) C B 9 k(O) (wk, r* + ro)

and hence

Vol 9 k(o) B 9 k(o) (wk, r* + ro) v 0

(r* +rot 2: (r* +rot·

Choose c: ~ ~ h~To)n. Since Qk-+ oo, we have A (c:) QJ;,^1 /

2

::::; r* + r 0 when

k is large enough. By the Bishop-Gromov volume comparison theorem we

have

Vol 9 k(o) B 9 k(o) (wk, A (c:) QJ;,^112 )

c -> ( A (c:) Qk,1/2 )n

Vol 9 k(o) B 9 k(o) (wk, r* + ro)

2: ( r* + ro )n 2: 2c:.

This is a contradiction and the lemma is proved. D

3.3. Second proof of the compactness of 11;-solutions - via

Proposition 20.6.

In this subsection we give another proof of Theorem 20.9, this time using

Proposition 20.6 (note that this result also depends on Proposition 20.4).

This variation on Perelman's proof is due to one of the authors [142]. Let

(Mk, 9k (t)) be a sequence in m-:;,~n and, given Xk E Mk and tk::::; 0, let

gk (t) = R^9 k (xk, tk) 9k (tk + Rgk (!k, tk))

be defined by (20.21). Again, the solution gk (t) is 'normalized' by the
property R 9 k (xk, 0) = 1.