1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

1. REDUCED VOLUME OF A STATIC METRIC 385

Let A (s) denote the volume of the geodesic (n - 1)-sphere of radius s cen-

tered at p. Since Reg 2': 0, we may apply the Bishop-Gromov volume com-

parison theorem (see (A.8)), which says that for s 2': r,

(8.8) A(s) ~ A(r) ;:=~ ~ n Vol!n(p,r) sn-l,

to estimate the second term on the RHS of (8.7):

where we made the change of variables rJ ~ 1;. Hence we have

V-(A ) ( 4 )-n/2 Vol B (p, r) n -n/^2 Vol B (p, r)^1

(^00) -rJ n-2 d
g, T ~ 7f / 2 + -
2
Tn 7f rn r2 e rJ^2 rJ
4,,-
Vol B (p, r) ( ( r^2 ) n/
2
= - + n -7f -n/21^00 e _,,, ''rJ n-2 2 d rJ ).
rn 41fT (^2) 4.,-r^2
(8.9)
This tells us that a lower bound for V yields a lower bound for the volume
ratios of balls. Note that
1 = r 1f-n/2e-lxl2 dx = nwn 1f-n/2 roo e-'TJrJ n;-2 drJ,
J~n 2 Jo
where Wn is the volume of the unit Euclidean n-ball. Hence
LEMMA 8.9 (The static reduced volume is bounded by volume ratios).

If (.Mn, g) is complete with Reg 2': 0, then for all r > 0 and T > 0,

(8.10) V (g T) ~ VolB (p, r) ((~) n/

2

+ ___!__).

' rn 41fT Wn

Thus, for r ~ p, the static reduced volume controls the volume ratio in
the sense that

(8.11) VolB(p,r) n 2': Cn -iv-(A g, p^2 ) '

r

where Cn ~ ( 2 .ft) n +Jn. Note also that if for some p EM and r > 0 we

have

(8.12) VolB n (p, r) <"" - '

r