1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

4. SOME RESULTS ON TYPE I ANCIENT SOLUTIONS

We conclude that for t < 0

(30.118) v~ (t) = r (- 47rt)-nf^2 e- et,(x,t)dμ,9oo(t) (x)

} Mt,

= (-47rt)-n/2 l L e-e~(x,t)-moo(y,s,t)dμh (y) ds

= (-47rt)- n/2 l et(f,;- a)2 L e-(y,t)dμh (y) ds

= (-47rt)-n/^2 l e.f.. JN e-(y,t)dμh (y)ds

=V;;(t).

This completes the proof of the claim.

By the claim, we obtain that

(30.119)

165

By the monotonicity of 11(; 0 ,o) (t), this implies that i{;o,O) (t) = const for t E

(-oo, 0). We conclude that g (t) is a gradient shrinker in canonical form with

C^00 potential function Rfxo,O) (t). By Proposition 30.32, (M,g (t)) is isometric to

(M~,g~(t)). D

The following result, obtained by X. Cao and Q. Zhang (see [51] for a proof),
ensures that some backward limit g~ (t) is nonflat.

THEOREM 30.34 (Existence of asymptotic shrinkers for Type I ancient solu-
tions). For any 1>,-noncollapsed complete Type I ancient solution of the Ricci flow,
there exists a backward limit which is a nonfiat shrinker.
Theorem 30.31 raises the following.

PROBLEM 30.35. Does there exist a backward asymptotic shrinker for any 1>,-
noncollapsed Type II ancient solution?

The answer to this question is yes when n = 3.

THEOREM 30.36. For any 3-dimensional 1>,-noncollapsed Type II ancient solu-

tion with bounded curvature, there exists a backward asymptotic shrinker.

PROOF. Let (M^3 ,g(t)), t E (-oo,O], be a 3-dimensional 1>,-noncollapsed Type

II ancient solution with bounded curvature. Then g (t) must have positive sectional
curvature (seep. 374 of [77]). By the argument of Proposition 9.29 in [77] (here M
may be either closed or noncompact since we have the 1>,-noncoll apsed assumption),
we have the following. There exists (xi, ti) E M x (-oo, 0) with ti -t -oo such
that (M^3 ,gi (t) ,xi), where

9i(t) ~Rig (ti+ Ri^1 t) and Ri ~ R 9 (xi, ti),

converges in the C^00 pointed Cheeger- Gromov sense to a noncompact 1>,-noncollapsed

steady GRS (M!o, g 00 (t), x 00 ), t E (-oo, oo ), with the property that R 900 achieves

its space-time maximum (of 1) at (x 00 , 0) and that Rm 900 > 0. Note that Rm 900

cannot have a zero because there a re no 2-dimensional nonflat 1>,-noncollapsed steady
solitons.