1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

6. IMPROVED VERSION OF NLC AND DIAMETER CONTROL 265

p EM, and r > 0, we have

( 6. 96 ) μ (g, r2) <log VolB (p, r) + 36 + lB(p,r) VolB (p, r) '

(

r2 f R+dμ)

• rn VolB(p,r) VolB(p,r/2)

where R+ ~ max { R, 0} is the positive part of the scalar curvature.

REMARK 6. 71. Proposition 6.64 follows from the above statement and

the Bishop-Gromov relative volume comparison theorem.

PROOF. We estimate each of the three terms on the RHS of (6.85) sep-

arately, where w is chosen by (6.86). Since 1¢'1 ::::; 3, IVdl = 1 a.e., and

supp ( ¢') c B (p, r) , the first term is estimated by

4r^2 JM1Vwl

2

dμ::::; 4 JM (41fr^2 )-n/

2

e-c1¢' (~) 1

2

dμ

<

36

VolB(p,r)

- VolB (p, r/2)'

where we used (6.94) to obtain the last inequality. We also have

r2 JM Rw2dμ = r2 JM R (47rr2rn/2 e-c¢ (~) 2 dμ

< r2 { R+dμ,

• VolB (p,r/2) lB(p,r)

where we used the fact that¢::::; 1 has support in B (p, r) and (6.94). Finally,
by Jensen's inequality (compare with (6.47)),

JM log (w^2 ) w^2 dμ 2: -logVolB (p, r)

since JM w^2 dμ = 1 and supp ( w^2 ) c B (p, r). Hence the third term has the
upper bound:

{ fw^2 dμ = { (-~log (47rr^2 ) - log (w^2 )) w^2 dμ::::; log VolB (p, r).

lM lM 2 rn

Summing the above three inequalities yields the proposition. D

Motivated by the expression on the RHS of (6.96), we define the maximal

function on a closed Riemannian manifold (JVt.n, g) by

MR(P, r) ~ sup Vi lB s2 ( )^1 R+ dμ.

O<s::Or 0 p, S B(p,s)

Since

lim s2^1 R+dμ = 0,

s-+0 VolB (p, s) B(p,s)