1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

208 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

w^1 ,^2 function by a c^1 function 1/J, then replace 1/J by 11/JI since their weak

derivatives are a.e. the same up to a sign, and then approximate 11/JI by a

nonnegative C^1 function with compact support.

For s > 0 we define

Ms ~ { x E M : 'ljJ ( x) 2 s} ,

F (s) ~ Vol_g (Ms).

Since 'ljJ has compact support, there exists ro < oo such that

Vol_g ({x EM: 'ljJ (x) > O}) = wnro = Vol(B (ro)),

where B (ro) C ~n is the ball of radius ro centered at the origin. Clearly
the function F : (0, oo) --+ [O, Wnr 0 ] is nonincreasing.
Let

h: ~n--+ ~

be a nonnegative rotationally symmetric function such that

(22.93) Vol({y E ~n: h(y) 2 s}) = F(s) ~ Vol_g ({x EM: 'ljJ (x) 2 s})

for all s > 0 and h (y) = 0 when IYI 2 ro. It is clear that there exists a

unique such function and that h (IYI) ~ h (y) is nonincreasing in IYI· We

define

M~ ~ {y E ~n : h (y) 2 s}

and r~ ~ 8M~. By definition, we have

Vol_g (Ms)= F (s) =Vol (M~)

for alls> 0. Since M~ is a round ball in Euclidean space, we have

(Area (r~)t =en (Vol(M~)r-

1

=en (Vol_g (Ms)r-^1

::::; ~: (Area_g(I's)t,

where we have used the isoperimetric inequality (22.101) below to obtain
the last inequality. We have shown for s > 0 that

(22.94) ( )

l/n

Area (r~) ::::; ~ Area 9 (rs)-

Since integration by parts holds for Lipschitz functions (see Lemma 7.113
in Part I), for any Lipschitz function A: [O, oo)--+ ~with .A (0) = 0 we have

(22.95)

{oo d.A {oo dF

Jo ds (s) F (s) ds = - Jo .A (s) ds (s) ds,

where we have used F (s) = 0 for s > supM 'lj.J. Applying the co-area formula
(see Lemma 5.4 in [45] for example)

{ H j V f j dμ = j

00

( { H da-) ds

JM -oo j{f=s}