1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

36 28. SPECIAL ANCIENT SOLUTIONS

STEP l. The cutoff function. First we improve the standard cutoff function
slightly. Given p E (2, oo], let 7J : JR ---t [O, l] b e a nonincreasing C^2 function such
that lrJ' ( u) I and lr7" ( u) I are bounded and

(28.2) 7) (u) = (2 - u)P if u E G, 2),

{

1 if u E (-oo, l],

0 ifuE[2,oo).

NotethatrJ(u) E [ 21 p,l] foruE [O,~). We have

7)^1 (u) = -p (2 - u)p- l ,

7)^11 (u) = p (p - 1) (2 - u?-^2

for u E G, 2), from which we may easily deduce that there exists const < oo such
t hat

(28.3a)

(28.3b)

for all u E [O, 2).

l'.=2.

irJ" ( u) I :S const · 7) ( u) P ,

(rJ' (u) )2 l'.=2.

---::; const · 7J (u) v

7J (u)

STEP 2. Defining the localization S of R. Now let (0, t 0 ) EM x (a, w) be any

point with R(O,to) < 0. Choose r 0 E (O,oo) so that

(28.4) Rc 9 (t) :S ( n - 1 )r 02 in Bg(t) ( 0 , r 0 ) for t E [a , t 0 ]

(clearly such an ro exists). Given any p > 0, define S: M x [a, to] ---t lR by

(28.5) S(x, t) = 7J C'(~,t)) R(x , t) ,

where

(28.6) r (x , t) ~ dg(t)(x, 0) + ~(n - l)r0-^1 (t - to).

Note that S has co mpact support. By (28.4) and Theorem 18.7(1) in Part III, we
have that for all x EM - Bg(t)(O , r 0 ) and t E [a,toJ,

(28. 7) ( :t - ~) r (x , t) ::=: o

in the barrier sense. From now on we shall assume that

(28.8) p :'.'.'. ro + 3(n^5 - l)r-1 0 (to - a).

STEP 3. The heat operator acting on S. We calculate

(28.9)