1547845447-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_IV__Chow_

106 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS

10. An unrescaled cigar backward Cheeger- Gromov limit

Let (5^2 , g(t)), t E (-oo,O), be a nonround maximal solution to the Ricci flow.
Recall from Proposition 29.36 that we may assume that v('lj;, e, t) converges on
52 - { N, S} to v 00 ('If;, B) = μ cos^2 'If;, whereμ > 0. Geometrically, this says that the
metrics g(t) converge pointwise on 52 - { N , S} to a flat cylinder metric as t --+ -oo.
In this section we show that g(t) pointed at N or S subconverges, after being
pulled back by conformal diffeomorphisms fixing N and S, to the cigar soliton as
t--+ -oo; see Proposition 29.38 below. Motivations for this result come from both
Theorem 29.2(2) and Lemma 29.4.
We consider g(t) pointed at N (at S is essentially the same). Recall that
u : 52 - { S} --+ JR^2 denotes stereographic projection and
g(t) = (u-^1 )*(g(t)) = u(t)geuc = v-^1 (t)geuc·

Let ti --+ -oo be any sequence of times. Define Ki ~ u-^112 (0, ti)· Note that by

(29.6) and v 00 (N) = 0, we have Ki --+ 0 as i --+ oo. Consider the sequence of
pulled-back solutions

(29.114)

where

(29.115)

Note that ui(O, 0, 0) = 1 for each i. By (29.7), we have that

(29.116)

is a solution to

(29.117)

We have the following.

PROPOSITION 29.38 (Backward convergence based at the poles to cigar soli-

tons). For any non round ancient solution there exists a subsequence such that the

functions vi(x, y , t) in (29.116) converge in C^00 on compact subsets of!R^2 x (-oo, oo)

to

(29.118) voo(x, y , t) =A (Beet+ (x - xo)^2 + (y - Yo)^2 ) ,

where A, B, C are positive constants and (x 0 , y 0 ) E IR^2. Hence (5^2 , g(t+ti), N) sub-

converges in the Cheeger- Gromov sense to a cigar soliton solution (IR^2 ,g 00 (t) , O),

where .9oo(t) ~ v~(t)geuc·

As in Lemma 29.4, we may interpret the sequence of solutions in (29.116) in the

context of Cheeger- Gromov convergence. Define the homotheties <I>i : JR^2 --+ JR^2 by

i(x, y) = (Kix, Kiy), which via stereographic projection correspond to conformal