1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 315

STEP 4. Finishing the proof of the theorem by jumping from L^2 to L^1.
The following argument is the parabolic version of an elliptic argument of
Li and Schoen [119]; see pp. 1269-1270 of Li and Wang [120].
For k E N U { 0} let
k
Pk ~ ro L 2-j = ro ( 2 - 2-k) ,
j=O

so that Pk is increasing in k, Po = ro, and limk-+oo Pk = 2ro. From (25.39)-

(25.40) with r = Pk and 'Y = 'Yk ~^2 --;,~~:

1
) (so that ryr = Pk+1) we obtain
fork~ 0

(25.41)

where (note that 'Yk - 1 = 2 k+~_ 2 )

A • ecs(l+'YkVKPk)i -(n~3)n A ( k+2 ) i

Ck =;= 1 C 0 Cn 2 - 2 < oo.

Pk (Vol_g B_g (xo, Pk))2

Note that since 'Yk ::::; 2 and ro ::::; Pk ::::; 2ro,

eCs(I+4VKro)i (n+s)n

ck ::::; 1 60 -^4 -cn2Ck+^2 ) i.

ro (Vol_g B_g (xo, ro) )2

(25.42)

We have the general inequality

llvllL2(Ppk+l) ::::; llvllii_

2

(PPk+1) llvll~~(PPk+l)'

so that (25.41) implies for k ~ 0

(25.43) llvllL=(PPk) ::::; Ck llvll~;(PPk+l) llvll~~(PPk+l).

This implies that for any R E N

llvllL^00 (Pro) = llvllL^00 (Pp 0 )

(25.44) <; m ( c. r) (Q M~~(P,.)) 11v11r;.'.(P,,).

Now

and