1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

52 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

Define fi ~ d(xi,8B(yi,1)) and take Ei E (0,1) to be chosen below. For
any x EB (xi, Eiri) we have d (x, 8B (yi, 1)) ~ (1 - Ei) fi so that

R (x)(l-Ei)^2 rt SR (x) d^2 (x, 8B (yi, 1)) SR (xi) rl.

Let

(i.e., Ei = ~(:~~) ), so that

R(x) S (1 +ci)R(xi)

(2) By (18.23) we have

that is,

R (x·) i r? i > - g? i R (y·) i = Et ( R (Yi) )2 -+ oo

(1 + Ei) 1 + Jl + Ei

using (18.22) and the fact that { Ei} is bounded.

(3) Since R (Yi) -+ oo, we have d (xi, 0) ~ d (Yi, 0) - 1 -+ oo as i -+ oo.

(4) This follows from (3) and ri s 1 by passing to a subsequence. D

2.3. Point picking when the change in R is unbounded.
The next result will be used in the proof of Perelman's compactness
theorem for A;-solutions with Harnack (see Theorem 20.9 below). The idea
of applying this result is to suppose that for a sequence of pointed solutions,
a uniform curvature bound at the basepoints does not imply a uniform
curvature bound in balls of fixed radii centered at these basepoints, i.e., one
does not have 'bounded curvatures at bounded distances'. One then wants
to derive a contradiction.

LEMMA 18.15 (Point picking for sequences of manifolds where the change
in R is unbounded). If (N'k, hk, Ok) is a sequence of complete^7 pointed Rie-
mannian manifolds with
Rhk (Ok)= 1

and if there exists a constant D > 0 such that

sup Rhk (w)-+ oo

wEBhk (Ok ,D)

ask-+ oo, then there exist Wk E Bhk (Ok, D + 1) and Sk E (0, D + 1) such
that

(i) Rhk (Wk) s~ _,; oo as k -+ oo and

(ii) for all w E Bhk (wk, sk)

Rhk (w) S 2Rhk (wk).

(^7) We just need the balls Bhk (Ok, D + 1) to be compact.