1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

186 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY

and for every (x, t) E P(xo, to, (cQ)-^112 , -(cQ)-^1 ) we have

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

Note that, assuming SUPMx[O,T] I Rm l(x, t) < oo, the above discussion

holds when the property J Rm J (x, t) ::::; 2Q is replaced by finding a point

(x, t) with J Rm J(x, t) ::::; Bl Rm J(x, l) for some e > 1. Later we will see that

a refinement of this method will yield (x, t) with other properties and we

can also drop the bounded curvature assumption (22.1).

1.1.3. Proof of Claim 1.

The following is the original Claim 1 on p. 24 of Perelman [152], which

is used in the proof of Theorem 21.9. Recall that Ma is defined in (21.16).

Claim 1 (Picking a well-chosen large curvature point). Let Eo >

O, a> 0, and A> 0. Suppose that (Mn,g(t)), t E [O,c^2 J, where EE (O,co],

is a continuous family of complete smooth Riemannian manifolds, xo EM,

and there exists a time ti E (0, c^2 ] and a 'nearby' point x1 E Bg(ti) (xo, co)

with large curvature:

a 1

(22.6) I Rm J(x1, tl) > - + - 2.

. ti Eo

Then there exists a 'not far' a-large curvature point (x, l) E Ma, with

(22.7) x E Bg(t) (xo, (2A + l)co) and l E (0, c^2 ],

such that we have curvature control:

(22.8)

for all

(22.9)

I Rm J(x, t)::::; 4J Rm J(x, t)

• 1 -

(x, t) E Ma with 0 < t::::; t and dg(t) (x, xo) ::::; dg(t) (x, xo) +Al Rm 1-2 (x, t ),

i.e., for all a-large curvature points not too much farther from xo than xis
from xo.^4
PROOF OF CLAIM 1. We shall define a sequence of points {(xk, tk)}k>l
with (xk, tk) E Ma and Xk E Bg(tk) (xo, (2A + l)co) inductively and show at
some finite step£< oo that (x, l) ~ (xe, te) is a point for which the claim is
true.
Let (x1, ti) be a point as in the hypothesis of the claim. Suppose

that, inductively, for some m E N the points (xk, tk) E Ma with Xk E

Bg(tk) (xo, (2A + l)co) and tk > 0 have been defined in the fashion described
below for 1 ::::; k ::::; m. Moreover, suppose that the point (x, l) in the claim
cannot be taken to be (xm, tm)· (If (x, l) can be taken to be (xm, tm), then

(^4) Note that in (22.9) the distance from x to xo is measured with respect tog (t) whereas
the distance from x to xo is measured with respect to g (t).
The point x is 'not far' from Xo and x1 whereas all we know (from the proof below)
about tis that it is in (0, ti].