1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

180 21. PERELMAN'S PSEUDOLOCALITY THEOREM

LEMMA 21.19 (Curvature gap for Type III solutions). If an eternal so-

lution (Mn,g(t)), t E (O,oo), on a compact manifold satisfies

(21.69)

CY

max IRml (x, t) :S -

xE.N! t

fort sufficiently large, where CY < 2 (n~l), then g (t) approaches almost fiat

in the sense of Gromov as t -+ oo.

PROOF. Throughout the proof we shall assume that t is sufficiently large

so that (21.69) holds. This curvature bound implies

tmax IRcl (x, t) :S (n - 1) a.

xE.N!

Since the length of a unit speed path/: [a, b] -+ M evolves by

:t L g(t) (t) = - i Re ( /

1

( s) , 1' ( s)) ds,

where ds is the arc length element, we have

d (n-l)CY

dtLg(t)(t):S t Lg(t)(/).

Since M is compact, this implies that the diameter satisfies

d (n-l)CY.

dt diam (g (t)) :::; t diam (g (t)),

so that

diam (g (t)) :S C tCn-l)a

for some constant C < oo. The inequality a < 2 (n~l) and (21.69) imply

diam^2 (g (t)) · ma;15 IRm (x, t)I :S C^2 CYt^2 Cn-l)a-l-+ O

XE;v<

as t -+ oo. This completes the proof by the definition of almost flat. D

§2. A remark about the form of Counterstatement A - (fix n and CY)

logically, the theorem is of the form: there exist o > 0 and co > 0 such that

P n,a ( o, co), where P n,a is a statement. Its negation is

(i) for every o > 0 and co> 0, not Pn,a (o, co).

This implies that

(ii) for every pair of sequences Oi-+ o+ and ci-+ o+, not Pn,a (oi,ci)·

This in turn implies that

(iii) there exist sequences Oi-+ o+ and ci-+ o+ such that not Pn,a (oi, ci)·

On the other hand, by the monotonicity property in Remark 21.10, we have
that (i) and (iii) are equivalent.

§3. Regarding the complete flat stationary solution

(M~,§oo (t), (x 00 ,0)), t E (-oo,O],

in (21.58), recall the following: