1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

174 21. PERELMAN'S PSEUDOLOCALITY THEOREM

to obtain a subsequence converging, in the C^00 pointed Cheeger-Gromov
sense, to a smooth limit solution

(21.42)

Since A = 100 ;e:oi --+ oo, this limit is complete and satisfies

(21.43) I Rm 9= I (xoo, 0) = 1

and

I Rm 9= I ( x, t) ::; 4 on Moo X (-~, 0 J.

Note that by the definition of C^00 pointed Cheeger-Gromov convergence

(slightly modifying Definition 3.6 in Part I) we have established the follow-

ing.

LEMMA 21.15. There exist

(1) an exhaustion {UihEN of M 00 by open sets with Xoo E Ui and

(2) a sequence of diffeomorphisms

(21.44) <l'>i : Ui --+ Vi ~ <l'>i (Ui) c Mi,

with <l'>i (x 00 ) = Xi and <l'>i (Ui) C B9i(o) (Xi, 1o),

such that the sequence

(ui, <Pi [gi(t)lv.D

converges in C^00 to ( M 00 , g 00 ( t)) uniformly on compact sets in M 00 x

(-~,OJ.

Now from Lemma 22.9 in the next chapter we have the following.

LEMMA 21.16. For the above subsequence of (21.38) converging to (21.42)

in the sense of Lemma 21.15, there exists a further subsequence such that

we have the convergences to certain C^00 limit functions

(21.45) Hi o <l'>i --t H 00 , Ji o <l'>i --t 100 , and Vi o <l'>i --t Voo

on M~ x (-~, 0) uniformly on compact sets in the C^00 -topology and we

have that H 00 = (-47rt)-n/^2 e-1= is a positive solution to the adjoint heat

equation

(21.46)

By Lemma 22.9, we also have

(21.47) Voo = (-t ( R9= + 2fl9=loo - l\79=lool

2

) +loo - n) Hoo::; 0

on M 00 x (-~,O), and on M 00 x (-~,O)

(21.48) D~iJ 00 = 2t IRc9= +\J9=\J9= loo+ ~g 001

2

2t - H^00

g=

::; o.