1547671870-The_Ricci_Flow__Chow

242 8. DILATIONS OF SINGULARITIES

which is the curvature bound we need for a subsequence to converge to a
Type I singularity model.
By classifying the limits of dilations of all sequences (xi, ti) chosen in
this manner, one can understand a Type I singularity at all fixed relative
scales in the interval (0, 1], where the relative curvature scale of a point
x E Mn at time t E [O, T) is defined to be the ratio

IRm (x, t) I

SUP Mn IRm (·, t) I"

There are certain properties common to any such limit.

PROPOSITION 8.20. Let (Mn,g (t)) exhibit a Type I singularity at T E

(0, oo), and let (xi, ti) be any globally curvature essential sequence. Then

any limit ( M~, g 00 ( t) ) is an ancient Type I singularity model that exhibits

a Type I singularity at some time w < oo.

PROOF. By the Type I singularity condition and Lemma 8.19, there
exist positive constants co < C such that

co c

-T :::; suplRm(-,t)I:::; --.

• t Mn T-t

Then by our choice of (xi, ti), there is c > 0 such that

c c

--T - ti -< IRm(xii · t)I i < - --T - ti.

The solutions (Mn, gi (t)) defined by (8.8) exist for -ai :::; t < wi, where

Cti =ti IRm (xi, ti)I > 0,

Wi = (T - ti) IRm (xi, ti)I > 0.

For any t E (ai,wi), the curvature Rmi of gi obeys the estimate

(8.12) __ c w· i_ :::; IRmi (x, t)I:::; __ C w· i _ _

C Wi - t C Wi - t

Indeed, one has

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

< T__ - t· i ---------C

• c T - ti - t IRm (xi, ti)l-^1
  C Wi

C Wi - t'

with the other inequality being obtained similarly.
Now since c:::; wi :::; C for all i, we can choose a subsequence (xi, ti) such
that w ~ limi--->oo Wi exists. Since Cti 2'. T I Rm (Xi, ti) I - C, we have ai -+ oo
by (8.3). So any limit (M~, g 00 (t)) is defined on (-oo, w). For each time
t E (-oo,w), there is It so large that t E (-ai,wi) for all i 2'. It, whence