1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

(^200) 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY
where a :::; 1. Given A 2: 67n, suppose that there exists (x, [) E M x (0, E^2 ]
with
(22.58)
(22.59)
dg(f) (x, x 0 ) :::; (2A + l)Eo,

• 
  
  
  
  • a
    Q-::?;= IRml(x,t) > t
  
  
  
  

and such that

(22.60) I Rm l(x, t) :::; 4Q

for all (x, t) E M x (0, f] satisfying

a - i

IRml(x,t) > t and dg(t)(x,xo):::; dg(f)(x,xo) +AQ-2.

Further assume that there exists

(22.61) t E [t -~Q-^1 , t]

such that

(22.62) 1 (- 177) v dμg(t) :::; -/31

Bg(t) :r,y t-t

for some /31 > 0. Let h be the cutoff function defined by (22.44) with b -::?;:.

lOEoA.

(1) Then

(22.63) JM vhdμg(t)'t=O:::; -/31 (1-~2) ·

(2) Let

H -::?;:. H h and J -::?;:. f - log h,

where h is the same cutoff function as above, so that

H(x,t)-::?;= (47r(t-t))-nl^2 e-f(x,t).

If we further suppose that A :::; 2 iJ-c:o and that^6

R(-,0) 2: -1 in Bg(o)(xo, 1),

then

(22.64) JM ( -fl\J fl^2 - j + n) fr dμg(O) ?_ /31 ( 1 - ~ 2 ) - ~ 2 - E^2 ,

where all the quantities on the LHS are evaluated at t = 0. Moreover

JM fr (x, 0) dμg(O) (x) :::; 1.

Note that supp (fr ( ·, 0)) c Bg(O) (xo, 20EoA) c Bg(o) (xo, 1).

(^6) Note that A ::'.'. 67n and A :::;
1 .·^20 ~^0 imply 1 that for the lemma to be nonvacuous, we
need co:::; 1340 n. In practice, we take A= lOOneo and so--+ 0 (see (21.20)).