1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

i7o 21. PERELMAN'S PSEUDOLOCALITY THEOREM

Let^8 (xii, lii) E Mia denote the bad points given by Counterstatement

A and Lemma 21.12. In particular, we have

dgi(tii) (Xii, Xoi) < C:Qi

and

a 1

I Rmgil(xii, tii) > - + 2·

tii c: 0 i

We apply the 'point picking' Claim 1 in §1 of Chapter 22 to the solutions

(Mr, 9i (t), xoi), t E [o, c:rJ,

1

(21.20) A--'---

i ...,.... lOOnc:oi'

and (xii, tii) E Mia as in Counterstatement A. Note that Ai --+ oo as

i--+ oo. Then we obtain 'well-chosen' ex-large curvature space-time points:

(21.21)

(21.22)

satisfying

(21.23)

and

(21.24)

for all (x, t) E Mia with

(xi, ti) E Mia,

Qi ~ I Rm gi I (xi,ti)

1

(21.25) 0 < t S 4 and dgi(t) (x, xoi) S dgi(ti) (xi, xoi) + AiQ~^2 •

Furthermore, by Claim 2 in §1 of Chapter 22 (which uses a< i 3 (n!i)vn)'

the (xi, ti) in (21.21) also satisfy

(21.26) I Rm gi I (x, t) S 4Qi

for all (x, t) such that

• a - i -

t· i - -Q-:- 2 i < - t < - t· i

That is, we have curvature control in backward parabolic cylinders centered

at (xi, ti) with relatively large spatial radii (since Ai--+ oo).

Note that since (xi, ti) E Mia, by (21.17) we have

• a

(21.27) Qi > 4'

in particular, ti - ~Qii > ti/2.

• 
  
  
  
  • i
    i.e., ti > aQi ,
  
  
  
  

(^8) The subscript 'l' is introduced here to mesh better with the notation of Claim 1 in
§1 of Chapter 22.