1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

210 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

Assuming this claim and using JM 'lj;^2 djl = fJRn h^2 dμJRn, we have

2:: o,

where we have used (22.89) to obtain the last inequality. Hence (22.92) is

proved modulo the claim.

Finally we give a proof of the claim. By the co-area formula, we have

Hence

(22.100)

Since h is rotationally symmetric and r~ is a round sphere centered at the

origin, we have by (22.94),

fr, j\7hj dCY]Rn •fr, J~hl dCY]Rn

s s

= (Area (r~) )

2

:S (;:)

2

/n (Area9(I's))^2

::::; (~)

21

n r lv'lj;/doc· r _

1

_

1

Ida:,

n Jrs Jr. \71/J

where we used the Holder inequality to obtain the last inequality. By this
and (22.100), we have