1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

3. TYPE I SOLUTIONS HAVE SHRINKER SINGULARITY MODELS 157

Generalizing the above discussion, we have

DEFINITION 30.21 (Set of Type I singular points). Let

(30.95)

~I ~ {p* EM : 3ti---+ w and Pi---+ p*' lim inf (w - ti) IRmgl (pi, ti) > o}.

i--+oo

That is, ~I is the set of points p such that there exists a sequence of (Riemann

curvature) Type I essential points approaching (p, w).

By (30.93), ~I=/=- 0 for any singular solution on a closed manifold.

Notwithstanding the possibility of (30.94), one can prove the following result,

which is a consequence of the above discussion plus pseudolocality.

THEOREM 30 .22 (Limits of Type I solutions with basepoints in ~I a re nonflat

shrinkers). If (Mn,g (t)), t E [0,w), is a Type I singular solution on a closed

manifold. Then for any p* E ~I, corresponding to Pj :::::: p* and any Aj ---+ oo, the

bounded curvature shrinker limit (M~,g 00 (t) ,p 00 ) in Proposition 30.19 is non-

Ricci fiat.

PROOF. Aiming for a contradiction, we suppose that g 00 (t) :::::: g 00 is Ricci flat

on M 00 for all t E (-oo,O). (This actually implies that (M 00 ,g 00 (t)) is isometric

to Euclidean space by Lemma 27.9.)

Let c:, 6 > 0 b e as in P erelman's pseudolocality Theorem 10.3 in [312] and

fix ro E (0, inj 900 (p 00 ) ). By the pointed Cheeger- Gromov convergence of gj (t) =

>..jg(w + >..j^1 t) to g 00 (t) and since g 00 ( - (c:r 0 )^2 ) is flat, we have that for J 2 Jo,

where Jo is sufficiently large,

1Rm91(-(c:ro)2) I:::; r()2 in B91(-(c:ro)2) (p*,ro)

and

Vol 91 (-(c:ro)2) B 91 (-(c:ro)2) (p*,ro) 2 (1-6)wnr 0.

By Theorem 10 .3 in [312] (see Proposition 1 in [211] by one of the authors for the

version we use), we have that for all J 2 Jo,^5

(30.96) JRm 91 J (x, t):::; (cro)-

2

in B 91 (-(c:ro)2) (p*,c:ro) x [-(c:ro)

2

, 0).

Since p* E ~ 1 , there exists ti ---+ w and Pi ---+ p* such that

c

(30.97) 1Rm 91 (pi, ti) 2 --

w - ti

for some c > 0. Fix a J 2 Jo· From (30.97), for i sufficiently large we have

(30.98) JRm 9 I (pi, ->..j (w - ti)) 2 >.. ( c ) 2 2 (c:ro)-

2

J j w - ti

and

->..j (w - ti) E [- (c:ro)^2 , 0)

since w - ti ---+ 0 as i ---+ oo. We obtain a contradiction to (30.96) since, for i
sufficiently large, we have
Pi E B 91 (-(c:ro)2) (p*,E:ro).

Hence g 00 (t), t E (-oo,O), is not Ricci flat. D

(^5) For the application of Proposition 1 in [211] it is not necessary that 9j (t) be d efined at
t = 0.