1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

5. NOTES AND COMMENTARY 37

we also obtain

Vol(g(t)) ~ (1-~,(g(O))t) n/

2

Vol(g(O)).

§2. (1) The following is used in the proof of Proposition 17.20. From

the formula

d~ (x-^8 logx) = x-^8 -^1 (1-b'logx)

for 8 > 0, we see that the maximum of x-^8 log x on (0, oo) is ie-^1 which

occurs at x = e^118. Hence

_! < w logw < __!_w^1 +8.

e - - 8e

Thus for any q > 0 and 8 > 0,

(17.115) J wq(logw)qdμ~-^1 - { wq(l+^8 )dμ+]_Vol(g).

M (8e)q JM eq

Note that we have for any p < n^2 :: 2

(17.116) JM r <p'P(log<p)Pdμ~ ( (n-2ne )-p r 2n

2 )p -e JM <p<n-2)dμ

1

+ eP Vol (g),

where we have used logx ~ lex^8 with 8 = (n~)P -1 > 0.

Following the proof of Lemma 6.36 in Part I, we conclude that for a > 0

1 1

2 2 2 n^4

(17.117) M <p (log <p) dμ 9 ~ a M IV 'Pig dμ 9 + ae (^40) s (M ,g )

+aVol(g)-^2 ln+

1

e^2 Vol(g).

(2) We can define Ui and <I>i in (17.52) explicitly as follows. Let inj (g)

denote the injectivity radius of g (note that inj (gi) = i~ --+ oo as i--+ oo ).

Choose orthonormal frames { e~} :=l at Xi with respect to 9i and let

'I/Ji : (JR.n, !J!Rn) --+ (TxiM, 9i (xi))

be the linear isometry taking the standard basis in JR.n to { e~} :=l' Let

U· i == • B (o ' inj(g)) yl2T;, c JR.n

and define i : Ui --+ M by