1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 151

Integrating (30.69) over [f, w] yields

dg(f) h'1(f), 1'2(f))

:S4B!(w-f)! (2nVM+jf(x 1 ,f)+~+jf(x 2 ,f)+~)

• 1

+ 2 const ( w - t ) 2.

Therefore estimate (30.63) follows since X1=1'1(f) and x2=1'2(f).

Finally, (30.64) follows from taking x 1 = p and x 2 = x in (30.63) and from

applying (30.59). D

Now we can apply the estimate above to the sequence used to define the reduced

distance £ 00.

PROPOSITION 30 .18. Under the hypotheses of Theorem 30.13 we have that for

any c E (0, w^00 ;°'^00 )

(30.70) d;oo(t) (x,poo) - c < f (x t) < cd;oo(t) (x, poo) + c

C (w 00 - t) -^00 ' - W 00 - t '

(30.71) IV fool (x, t) :SC ( Jw~ _ t + dg::) ~~ x)),

(30.72)

I

f)foo I (x, t) :SC (d;oo(t) (p, ~) + 1 )

fJt (w 00 - t) Woo - t

on M 00 x (a 00 + c, w 00 ) for some C < oo. Analogous estimates hold for a reduced

distance based at the singular time of a single Type I solution.

PROOF. Applying inequalities (30.64) and (30.47) to (Mf,gi (t) ,pi), t E [ai, wi],

in the hypothesis of Theorem 30 .10, we have that for a ny (x, t) E M 00 x [a 00 +c, w 00 )

d~;(t) (x,p 00 ) _ Cd~;(t) (x,p 00 )

C ( ) - C :::; fi(x, t) :::; + C ,

Wi - t Wi - t

where C < oo is independent of both (x, t) and i and where i is sufficiently large.

Hence, taking the limit as i-+ oo, we obtain (30.70).

R ecall from Theorem 30.lO(i) that £ 00 is locally Lipschitz. By applying (30.60)

and (30.61) to fi defined in Theorem 30 .10 and then taking the limit of ii, we obtain

(30.71) and (30.72), respectively. D

Now we can give the

PROOF OF THEOREM 30.13. (1) V 00 (t) :::; l. For each i the reduce d volume
of the solution gi (t) based at (pi,wi ) is

(30.73)

Since each solution gi (t), t E [ai,wi], is complete with bounded curvature, by
Lemma 8.16(ii) in Part I we have that each Vi (t) is integrable. Moreover, by
Theorem 8.20 and Corollary 8.17, both in Part I, we also have that each Vi (t) is