1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

184 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

1.1. Point picking method for points with a-large curvature.

In this subsection (Mn,g(t)), t E [O,T], shall be a smooth family of

complete Riemannian manifolds with

(22.1) sup IRml < oo.

Mx[O,T]

1.1.1. Defining small and large curvature points.

DEFINITION 22.1 (a-small and a-large curvature points). Given a E

(0, oo), we say that a point (x, t) is an a-small curvature point if

(22.2) IRml(x,t) :St;· a

On the other hand, recall from Definition 21.11 that (x, t) is an a-large

curvature point if

(22.3)

a

IRmj(x,t)>-.

t

Note that inequality (22.2) is scale invariant in the following sense. If

ti Rm 9 l(x, t) :::; a and CE (0, oo), then ti Rm 9 c l(x, t) :::; a, where gc (i) ~

Cg ( t) and i ~ Ct.

Note also that for an expanding soliton with bounded curvature and

singular initial time 0, we have supM IRmg(t) I = co~st.

For (x', t') EM x [O, T], we define the parabolic cylinder

P(x', t', r', TJ) ~ { (x, t) : dg(t') (x, x') < r', t E [t', t' + TJl}

= Bg(t') (x', r') x [t', t' + TJ].

Here r' > 0, whereas TJ E JR; our notation is such that [a, b] denotes the

closed interval of real numbers between a and b, where we allow b < a.

1.1.2. Finding large curvature points with local curvature control.

Now fix s E (0, 1) and let (xo, t 0 ) EM x [T/2, T] be a point with 'large

enough' norm of curvature (relative to the length of the time interval), i.e.,

(22.4) Q ~ IRml(xo,to) >Ts·^8

Assume that (xo, to) is an a-large curvature point, i.e., I Rm l(xo, to) > ~­

Let H ~ fQ. We claim that we can find another 'better chosen' a-large

curvature point (xo, to), with io 2: t' such that

A A 8

Q ~ I Rm l(xo, to) 2: Ts

and such that for every (x, t) E P(x 0 , to, (sQ)-^112 , -(sQ)-^1 ) we have the
local curvature bound

(22.5) I Rm l(x, t):::; 2Q.