1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. EXISTENCE OF A MINIMIZER FOR THE ENTROPY 27

REMARK 17.27. Note that (17.77) is the same as (17.15) with r = 1. By

scaling, one sees that the lemma holds for solutions of (17.15) with arbitrary

T > 0.

The proof is via a monotonicity formula in the radial direction emanating

from p. Denote by S (p, r) the geodesic sphere of radius r centered at p. Let

J (0, r), where 0 E sn-l (1) and r E (0, inj (p)), be the Jacobian of the

exponential map in spherical coordinates, so that

dμ (expp (rO)) = J (O,r) (expPL (d0"5n-1(1)) /\ dr,

where sn-l (1) C TpM is the unit sphere and d0"5n-1(i) is its volume (n - 1)-

form. Note that lim __..!_ 1 = 1.

r-tO rn-

Now define

F: (0, inj(p)) ~JR

by

f s(p,r) Woo dO" fsn-l(I) Woo ( 0, r) J (0, r) dO"sn-l(I)

F (r) = J S(p,r) dO" = J sn-l(I) J (0, r) d0"5n-l(I) ,

where dO" denotes the induced volume (n - 1)-form on S (p, r).^10 Since

w 00 (p) = 0 and w 00 is continuous, we have

lim F (r) = 0.

r-to+

We shall derive a differential inequality for F (r) to show that F (r) = 0 for

r sufficiently small. Since w 00 2:: O, this implies w 00 = 0 in a neighborhood

of p.

If we set

J S(p,r) BwBr^09 d O' J sn-l(I) Bwoo(B,r) Br J (ll u, r ) d 0"5n-^1 (l)

G(r) = = J ,

fs(p,r) dO" sn-1(1) J (0, r) dO"sn-l(l)

then

F' (r) = fs(p,r) ~ dO" + fs(p,r) woo fr log J dO"

f s(p,r) dO" f s(p,r) dO"

f s(p,r) Woo dO" f s(p,r) fr log J dO"

(f s(p,r) dO")

2

(

B f ( ) .E.. log J d<Y )

J S(p,r) Woo -Br log J - s p,r f s(p,r) ar d<Y dO"

= G(r) + f d

S(p,r) O'

Since

{) J ..§__ log J dO"

• 1 og J - S(p,r) Br _ - O ( ) r ,

or f s(p,r) dO"

(^10) That is, F (r) is the average value of Woo on S (p, r).