1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

168 21. PERELMAN'S PSEUDOLOCALITY THEOREM

(Mi,gi(t),xoi), t E [O,cr], where Ci E (O,coi], for i EN, of the Riccifiow
with bounded curvatures such that

(21.12)

(21.13)

Rgi (x, 0) ~ -1 in Bgi(o) (xoi, 1),

(Areagi(o)(an)t ~ (1 - oi)cn (Volgi(o)(n)r-

1

for all regular domains n c Bgi(o) (xoi, 1), and there exist 'bad' points and

times Xi E Mi and ti E (0, ctJ satisfying

(21.14)

and

(21.15)

for all i EN.

In view of the above counterstatement, we make the following:

DEFINITION 21.11. Given a continuous family of complete smooth Rie-

mannian manifolds (Mn, g (t)), t E [O, c^2 J, where c > 0, and given a E

(0, oo), let

(21.16) Ma 7. { (x,t) EM x (O,c]:^2 IRml(x,t) > t a}

be the set of a-large curvature points. ,

Hence for (Mi, gi (t)) and a as in Counterstatement A, the set

(21.17)

is nonempty for all i.
We shall show that Counterstatement A leads to a contradiction, from
which Theorem 21.9 follows.

2.2. Adjusting the bad sequence of points and times - getting

local curvature control.

For later use in carrying out the proof of Theorem 21.9 (see Lemma
22.13, in particular (22.57), below), we observe the following.

LEMMA 21.12 (An adjustment for the sequences ci and (xi, ti)). For each

i E N, we may assume, by adjusting ci > 0 smaller if necessary, that

(21.18)

a 2
I Rm gJ (x, t) ~ - + 2 whenever t E (0, ct] and dgi(t) (x, Xoi) ~ coi,
t coi
and we may also adjust (xi, ti) accordingly, so that Counterstatement A still
holds for the adjusted ci and (xi, ti), for all i E N.^7

(^7) Note that we adjust neither Xoi nor Oi.