1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

5. NO FINITE TIME LOCAL COLLAPSING 261

since r^2 R::::; ci(n) in B(p,r), supp(w) c B(p,r), and JMw^2 dμ = 1. By

(6.86), we have a.e.,

r2 j\7wl2::::; (4wr2rn/2 e-c [¢'[2::::; 9 (4w)-n/2 e~c

r

9 (4w)-n/2 eG3(n,p)

< - VolB(p,r ) '

using WI ::::; 3, j\7dl = 1 a.e., and (6.88). Since [\7wl has support in B (p, r),
we have

JM 4r2 j\7wl2 dμ::::; 9 (4w)-n/2 eG3(n,p)_

We conclude that the energy part of the RHS of (6.85) is bounded from
above:

JM r^2 ( 4 [\7w[^2 + Rw^2 ) dμ::::; c1 (n) + 9 (4w)-n/^2 eG^3 (n,p).

Now we consider the entropy part. Since f = c - log ( ¢^2 ) , by ( 6.87), we
have

JM fw^2 dμ ::::; JM ( c - log ( ¢^2 )) w^2 dμ

= c - ( 4wr^2 )-n/^2 e -c JM log ( ¢^2 ) ¢^2 dμ.

Since -xlogx::::; 1/e and¢ has support in B (p,r), we have^17

JM log (¢^2 ) ¢^2 dμ;:::: -~ VolB (p,r).

Hence

{ fw2dμ::::; c + ~ (4w)-n/2 e-cvolB (p, r)

JM e rn

::::; c + ~ ( 4 w)-n/2 eG^3 (n,p)

e

::::; log Vol B (p, r) + ~ ( 4 7r )-n/2 ec 3 (n,p).

rn e

We conclude that

μ (g,r A^2 ) :S:log VolB(p,r) n +C2 ( n,p, )

r

where C 2 (n, p) ~ (9 + ~) (4w)-n/^2 eG^3 (n,p) + c1 (n). This proves the first
part of the proposition.

l 7 Our convention is 0 · log 0 = 0 since lime-->D clog c = 0. Also, log ( ¢?) = 0 in
B (p, r/2), so in fact JM log(¢?) q?dμ ::'.': -~Vol (B (p, r) - B (p, r/2)).