1547671870-The_Ricci_Flow__Chow

5. DILATIONS OF INFINITE-TIME SINGULARITIES 249

If (8.25) holds for some c > 0, there exists a time Tc: < oo such that for all
t 2: Tc: we have tsupMn IRm (-, t)I :S 2c. In particular,

Cc

sup !Re(-, t)I :S - ,

Mn t

for all t 2: Tc:, where C depends only on n. Hence

dt dL (T) :'S \dL dt (T) I :'S Cc --;-L (T)

for T 2: Tc:. Integrating this inequality from time Tc: to time t > Tc: implies

L (t) :SL (Tc:) ( ;c:) Cc:= L (Tc:) Tc:-Cc:. tee:.

In particular, if we choose 0 < c < 1/ (2C), then any two points in (Mn, g(t))

can be joined by a path of length

L (t) :S [diam (Mn' g(Tc:) )] yc:-Cc:. tl/2-o

where o ~ 1/2 - Cc > 0. This implies (8.26) and proves the claim. D

Because 3-dimensional nilmanifolds are known to be geometrizable, we

now describe how to obtain Type III limits under the assumption a > 0.

By definition of a, there exist sequences (xi, ti) with ti / oo such that

ti !Rm (xi, ti)I ~ ai---+ a.

Choose any such sequence. Also by definition of a, there is for any c > 0 a

time Tc: E [O, oo) such that

tlRm(x,t)I :S a+c

for all x E Mn and t E [Tc:, oo). The dilated solutions 9i (t) exist on the

time intervals [-ai, oo) and satisfy

if

that is if

IRmi(x,t)I = IRm(~i,ti)l \Rm(x,ti + IRm(~i,ti)I)\

I Rm ( x, ti+ IRm(~;,t;)I) I (ti + IRm(~;,t;)I)

!Rm (xi, ti) I ti + t

a+c

<--

• ai + t'

t 2: !Rm (xi, ti) I (Tc: - ti).

Note that for any fixed c > 0, we have

!Rm (xi, ti)I (Tc: - ti)---+ - a

and

--a+c ---t a+c --

ai + t a + t