150 30. TYPE I SINGULARITIES AND ANCIENT SOLUTIONSin Part III with K = w"'!_t and radius = K -^1 /^2 , we have
(30.66) ( () ) ( ( ) ( )) lO(n- 1) r;::; lO(n-l) Af1
/2
- 8
d 9 ct) 'Y1 t , ')'2 t ::; - 3 -v K = - 3 - 1.
- 8
t (w-t)2
We can bound the term on the second line of (30.65) by using the equalityl'V adg(t) lg(t) = 1 as well as (7.54) in Part I. Namely, we have
I (\7 adg(t) hl (t), ')'2 (t)), ')'~ (t)) I ::; h~ (t)lg(t)
= l'Vfl 9 (t) ha (t), t)< A+ Bf h a (t), t)
w-twhere we have used (30.56) in the second inequality. By combining the above, we
obtain(30.67)
+ ~ A+ Bf ha (t), t)
L w-t
a=lSince 'Ya is a minimal £-geodesic, for any t E (f, w) we have(30.68)
since IRlmax (£) ::; ':_,
2
j. Hence, by applying (30.68) to (30.67), we have for any
t E (f,w)
(30.69) -! ( dg(t) hl (t) , ')'2 (t)))
Bl/ 2 (w - f)l/ 4 ( / A / A )
::; (w-t) 314 2nvfM+yf(x1,f)+ B+yf(x^2 ,t)+ B
const
+ 1 >
(w - t)2