1547671870-The_Ricci_Flow__Chow

5. DILATIONS OF INFINITE-TIME SINGULARITIES 247

Now ai --+ oo, but we do not know that Wi also tends to infinity. This
motivates finding a better sequence of points and times (xi, ti)·
Refining our choice by the method of Section 4.2, we pick points (xi, ti)
so that

ti (Ti - ti) !Rm (xi, ti)I ~ 1 _Di--+1.

supMnx[O,T;] [t (Ti - t) !Rm (x, t)i]

This guarantees that O:i --+ oo and Wi --+ oo. Indeed, it follows from (8.21)
that

1 o:iwi ti !Rm (xi, ti)! (Ti - ti)

a-:-i^1 + w-:-i^1 O:i + wi Ti

1

T

sup t IRm (x, t)I (Ti - t)

i Mnx[O,Ti]

1

2: - sup t !Rm (x , t)I --+ oo,

2 Mn x [O,T;/2]

hence that o:i^1 --+ 0 and wi^1 --+ 0. Then as in (8.22), we get

IRmi (x, t)I

(ti+ IRm(;;,t;) I) (Ti - ti - IRm(;;,t;) I) \Rm (x, ti+ IRm(;;,t;)I) I

ti (Ti - ti) !Rm (xi, ti) I

ti IRm (xi, ti)I (Ti - ti)

x -----------------~

(tilRm(xi, ti)I +t) (Ti-ti-IRm(;;,t;)I)

<--^1 x-O:i ---Wi

• 1 - Di O:i + t Wi - t
  for all x E Mn and t E [-o:i,wi)· Since Di --+ 0, O:i --+ oo, and Wi --+ oo,
  we conclude that the pointed limit solution (M~, g 00 (t), x 00 ), if it exists,
  is defined for all t E ( -oo, oo) and satisfies the curvature bound

sup IRmool :S 1 = IRmoo (xoo,O)I.

M;;,x(-00,00)

5.2. Type III limits of Type III singularities. Recall that a solu-

tion (Mn, g (t)) of the Ricci flow forms a Type III singularity at T = oo if

the solution exists for t E [O, oo) and satisfies

sup t !Rm(-, t)I < oo.

Mnx[O,oo)

Define

(8.23) a~ limsup (tsup IRm (-, t)I) E [O, oo).

t-->oo Mn

The nonnegative number a is analogous tow defined in equation (8.14). We

shall now demonstrate that one may assume a is strictly positive. We begin