1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

2. REDUCED VOLUME AT THE SINGULAR TIME FOR TYPE I SOLUTIONS 147

Therefore, applying this and the Type A assumption to (30.50), we obtain

(30.51)

IK (r)I

< i w(w _ i) ~ ( Cn,c,M + Cn,c,M 1 1' (i) I + nM Ir' (i) 1

2

+ n

2

M )di

• t (w-i)^2 r (w-i)~r (w-ir (w-ir+^1

i

w ( ' ) ~ ( 2Cn,c,M (nM + Cn,c,M) 1 1' (i) 1

2

n^2 M ) d '

< W - t 2 + , r + +1 t

• t (w-i) r (w-t) (w-ir

< i w , ~ (nM + Cn,c,M) ( R + 1 1' (i) i2) ,

_ ( w _ t ) ( , r dt

t w - t

+iw (w-i)~ (2Cn,c,M+(nM+Cn,c,M)n

2

M + n

2

M )di

t (w-i)2r (w-ir+1

~ (nM + Cn,c,M) i w (w - i) ~-r ( R +Ir' (i) i2) di

2CncM+(nM+CncM)n^2 M( ) 2-2r n^2 M( )2-r

+ " 5 " w-t2 +-3-w-t2'

2 - 2r 2 - r

provided r < ~.
Now we consider the specia l case where r = 1 (i.e., we have a Type I singular
solution) since this gives us a clear way to estimate the penultimate line of (30.51).
In this case we have

(30.52)

1

+ 2 (nM + Cn,c,M) (w - t)^2 R. (z, t).

STEP 2. Bounding IV'R.I. Again using r = 1, by (30.49) and (30.52), we have

(30.53) IV'R.^12 (z, t) ~ IRI (z, t) + IK (r)I + R. (z, t)

(w - t)3/2 w - t

< A+ BR. (z, t)

• w-t

on M x [a+ c,w), where

(30.54)

(30.55)

A~ 4Cn,c,M + 2 (nM + Cn,c,M + ~) n^2 M,

B ~ 2 ( nM + Cn,c ,M) + l.

That is, we have the key inequality:

(30.56)

on M x [a+c,w).

l\7£1 (z, t) ~ J A+ BR. (z, t)

w-t