1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

260 6. ENTROPY AND NO LOCAL COLLAPSING

where d 9 ( x, p) is the distance function and the constant c = c ( n, g, x, r)

is chosen so that JM w^2 dμ = 1. Note that w is a Lipschitz function and

definition (6.84) implies

(6.87) f = c - log ( </>^2 ).

(We abuse notation and write</> (x) = </> ( d^9 ~,p)) .) The constant c is related

to the volume ratio Vol~~p,r) by the following.

LEMMA^1 6.65. There exists C3 (n, p) < oo such that for any r E (0, p),

(6.88) 1 og VolB(p,r) n - C 3 ( n, p ) < < c 1 og VolB(p,r) n.

r r

Proof of the lemma. (1) Since JM w^2 dμ = 1,

(6.89) i= (47rr^2 )-n/

2

JM<1>(d(~,p))

2

dμ(x).

Applying </> ~ 1 and supp ( w) c B (p, r) , we have

ec ~ ( 47rr^2 )-n/^2 Vol B (p, r) ,

which implies

(6.90) 1

VolB(p,r)

c ~ og n.

r

(2) On the other hand, since </> = 1 on [O, 1/2] , by (6.89) we have

c -> --n 2 1 og ( 4 7r ) + 1 og VolB rn (p,r/2).

Since Re 2: -ci ( n) r-^2 in B (p, r) and r ~ p, by the Bishop-Gromov relative

volume comparison theorem, there exists C4 (n, p) < oo such that

VolB (p, r) ~ C4 (n, p) VolB (p, r/2).

Thus

(6.91) c > _ -C 3 ( n, p ) + (^1) og VolB(p,r) n ·
r
This completes the proof of the lemma.
To prove the proposition, we estimate the two terms
JM r^2 ( 4 J\7wJ
2

• Rw^2 ) dμ +JM fw^2 dμ = K (9, w, r^2 ) + n
  on the RHS of (6.85) separately. First we have
  JM r
  2
  Rw
  2
  dμ ~ c 1 (n)