1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS 161

Since f (-1) = f (-1), we conclude that }(t) = f (t). From (30.105) we conclude

(30.102).

(2) The "if" part of the lemma follows from Theorem 4.1 in [77]. D

EXERCISE 30.30. Show that if f is normalized so that R + IV fl^2 - f = 0 at

t = -1, then

fort E (-oo, 0).

1

R + IV !1^2 + -f = 0

t

SOLUTION. One computes that

:t ( R +IV !1

2

+ ~ f) = -~ ( R +IV !1

2

+ ~ f).

Analogous to P erelman's "asymptotic shrinker" result for K:-solutions (see The-

orem 28.24), we have the following application due to Naber [273] of the reduced

volume monotonicity based at the singular time.

THEOREM 30.31 (Backward and forward asymptotic shrinkers for K:-noncollapsed

Type I ancient solutions). Let (Mn, g (t)), t E (-oo, 0), be a K:-noncollapsed (below

all scales) ancient so lution satisfying

c

IRml (x, t) :::; ltf on M x (-oo, 0)

for some C < oo. Then, for any xo EM and sequences Tt ')i 0 and Ti- /' oo, there

exist subsequences such that (Mn,gf (t), (x 0 , -1)), defined by (30.101), converges

to a complete shrinking GRS

((M~r ,g! (t), (x~, -1)).

Furthermore, the forward asymptotic soliton (Mt,, gt, (t)) and the backward

asymptotic soliton (M~, g~ ( t)) are K:-noncollapsed, are in canonical form, but

are possibly isometric to Euclidean space.

PROOF. By the Type I hypothesis, we have

I I

± ± c

Rm 9 ; (x, t) =Ti 1Rm 91 (x, Ti t):::; ltf

on M x (-oo, 0), for a ll i. Since g (t) is K:-noncollapsed for some K: > 0, by the scale-

invariance of this property, we have that the g'f= (t) are all K:-noncollapsed for this
same K:. Therefore we may apply Hamilton's Cheeger-Gromov-type compactness

theorem to conclude that there exists a subsequence such that (M, g'f= (t), (xo, -1))

converges to a complete limit solution

((M~t,g!(t),(x~,-l)), tE (-oo,O),

where g~ (t) is K:-noncollapsed and where 1Rm 9 ~ I (x, t) :::; 1 ~. In particular, there

exist smooth embeddings

f=: (ui±,x~) c (M~,x~)-+ (M,xo),