1547671870-The_Ricci_Flow__Chow

244 8. DILATIONS OF SINGULARITIES

hence if

t E [-IRm(xi,ti)I (ti - tE:) ,wi)·

Note that Wi _____, w and that

i---->oo .lim IRm (xi, ti)I (ti - tE:) = oo

for any E > 0. So for a pointed limit solution (M~, g 00 (t), xoo) of a sub-
sequence of (Mn, gi ( t) , Xi), we can let i ____, oo and then E ~ 0 in estimate
(8.16) to conclude that
w
IRmoo (x, t)I :S w t

for all x EM~ and t E (- oo,w). Because IRmi(xi,O)I = 1 for all i, the

limit satisfies IRmoo (xoo, O)I = 1.

4.2. Type II limits of Type Ila singularities. An important source

of intuition into the expected behavior of Type Ila singularities is given by

the conjectured degenerate neckpinch described in Section 6 of Chapter 2.

Recall that the Type Ila condition says that

sup JRm (-, t)I (T - t) = oo.

Mnx[O,T)

This will allow us to select a sequence of points and times so that the di-

lated solutions have uniformly bounded curvatures on larger and larger time

intervals both forwards and backwards in time.

First choose any sequence of times Ti / T. Given any c E (0, 1), let

(xi, ti) be a sequence of points and times such that ti _____, T and

(8.17) IRm (xi, ti)I (Ti - ti)~ c sup IRm (x, t)J (Ti - t).

Mnx[O,Ti]

In other words, if we were doing Type I rescalings relative to the intervals

[O, Ti], we would choose the curvatures at (xi, ti) to be comparable to their

maxima on Mn x [O, Ti]· Since Ti < T, such points and times always exist.

When Mn is compact, they also exist for c = 1.

Consider the dilated solutions given by (8.81. Condition (8.17) implies

that for

we have