1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

3. SPACE-TIME POINT PICKING WITH RESTRICTIONS 57

nonnegative sectional curvature and 0 EM, then there are at most a finite

number of disjoint balls B (p, r) with

E

sect (g) 2': 2 in B (p, r)

r

(called curvature c:-bump) and

d (p, 0) 2': Ar

(called A-remote).

3. Space-time point picking with restrictions

Besides spatial location, as considered in the previous section, in Ricci
flow we also have temporal location. In this section we discuss space-time
point picking techniques which we shall apply in §2 of Chapter 20 when
discussing almost 11:-solutions.

Let (Nn, h (t)), t E [to, OJ, be a (not necessarily complete) solution to

the Ricci flow with nonnegative Ricci curvature. Let B and C be positive
constants and define

(18.26) NB,c ~ { (y, t) EN x (to, OJ : R (y, t) > C + B(t - t 0 )-i}.

We call NB,c the set of 'large curvature points'.

Let p EN and suppose the metric ball Bh(o)(P, 1) is compactly contained

in N. Assume there exists (yi, ti) E Bh(t 1 )(P, ~) x (to, OJ which satisfies

(18.27) R(yi, ti) > C + B(ti - to)-1,

i.e., (yi, ti) E NB,C·

Let (}' E (0, 1) and

(18.28) Ai((}') ~ ( C + B(ti - to)-i) ·min {~(ti - to), 23

1

04 }.

LEMMA 18.24 (Existence of a large curvature point with local control).

Let (Nn, h (t)) with Re 2': 0, p E N, (yi, ti), and Ai((}') be as above. If

Ai((}') > 1, then the following property holds. For any A E (1, Ai((}')] there

exist t E (to, ti] and y E Bh(t) (p, i) such that t - to 2': (1 - (}')(ti - to),

R (y, t) > C + B(t* - to)-1,

and

(18.29)

for those points (y, t) E NB,C which satisfy

(18.30a)

(18.30b)

t E (t* - AR-i(y*, t*), t*],

YE Bh(t) (p, dh(t) (y,p) + Ai/^2 R-if^2 (y, t)).