1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

266 6. ENTROPY AND NO LOCAL COLLAPSING

the quantity MR(P, r) is a well-defined finite number for 0 < r < oo. Clearly,

if ri :::::; r2, then MR(P, ri) :::::; MR(P, r2)· By (6.96), we have

(6.97) μ (g, A s^2 ) :::::;^1 og VolB(p,s) sn + + ( 36 M R ( p, s )) VolB VolB(p,s) (p, s/2) ·

Interestingly, the factor v~~k~;:;/~) on the RHS above does not prevent one

from estimating volume ratios byμ and MR only under a scalar curvature

upper bound. We shall prove the following.

PROPOSITION 6. 72 (Bounding volume ratios by Vr and MR). If (ktn, fJ)

is a closed Riemannian manifold and 0 < s:::::; r, then

where

Vr (g) ~ inf μ (g, r) 2': v (fJ).

TE(O,r^2 ]

In particular, if R:::::; c1 (n) r-^2 in B (p, r), then MR (p, r):::::; c1 (n), so that

REMARK 6.73. Given p EM, the function r f--+ e_3n36evr(9)e-3nMR(p,r)

is nonincreasing.

PROOF. If v~~kB~~;/~) :::::; 3n (in this case we say that at p the volume

doubling property holds at scales), then (6.97) implies

and the estimate follows.

If VolB(p,s/VolB(p,s) 2 ) _ > 3 n , th en smce. (^1) Imk-+oo. VolB(p,s/Vo1B(p,s/2k) 2 k+l) = 2 , t n h ere exists.
VolB(p,s/2k) n VolB(p,s/2i) n.
k EN such that VolB(p,s/ 2 k+l) :S 3 and VolB(p,s/ 2 i+l) > 3 for all 0 :Si< k.
Applying (6.97) to B (p, s/2k) , we get
( (
μ f;, s/2 k)2) :::::; log VolB (s/(p, s/2k) ( ( k))
2 kt +^36 +MR p, s/2. 3n.