1547671870-The_Ricci_Flow__Chow

4. DILATIONS OF FINITE-TIME SINGULARITIES 243

it follows from (8.12) that (M~, g 00 (t)) exhibits a Type I singularity at
w E [c, C]. 0

REMARK 8.21. A consequence of the results in Chapter 6 is that any

solution starting at a compact 3-manifold of positive Ricci curvature en-

counters a Type I singularity. Furthermore, any Type I limit formed at that

singularity will be a shrinking spherical space form.

We can satisfy the stronger condition (8.1) by being more careful in

picking the sequence of points and times. In particular, a necessary but not

sufficient condition is that the relative scales of the points (xi, ti) tend to 1

in the sense that

(8.13) lim IRm (xi, ti)I = 1.

i--+oo sup Mn I Rm (ti) I

Note that when Mn is compact, we may choose Xi so that

IRm (xi, ti)I =max Mn IRm (-, ti)I,

but this is not necessary (and not always possible when Mn is noncompact).

Define

(8.14) w ~ limsup IRm (-, t)i (T - t).

t/T

By Lemma 8.19 and definition (8.11), we have w E (0, oo). Take any se-

quence of points and times (xi, ti) with ti / T such that

(8.15) IRm (xi, ti)I (T - ti)~ wi-+ w.

Assuming (8.13) holds, this condition is equivalent to the requirement that

the times ti satisfy

sup IRm (-, ti)I (T - ti)-+ w.

Mn

In other words, we want the Type I rescaling factors to tend to their lim sup.

Consider the dilated solutions 9i(t) of the Ricci flow defined by (8.8).

By definition of w, there is for every E > 0 a time tc E [O, T) such that

IRm (x , t) i (T - t) ::::; w + E

for all x E Mn and t E [tc, T). The curvature norm of 9i then satisfies

(8.16)

if

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

\Rm ( X, ti+ IRm(;;,t;)I) I ( T - ti - IRm(;;,t;)i)

IRm (xi, ti)I (T - ti) - t

w+E

<--

- Wi - t'