90 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
A consequence of the above proposition is the following.
LEMMA 29.23 (Areas of the two isoperimetric regions are comparable). Thereexists a constant c > 0 such that for any minimizer "ft of I ( · ; g( t)) the areas of the
isoperimet'ric regions are bounded from below by
(29.72) cltl ::::; A ±('Yt)::::; 87rltl fort E (-oo, -1].
PROOF. By Proposition 29 .22, we have thati i i 47r L;C;l bt)
A± bt) ::::; A+ bt) +A= bt) ::::; 1 + c lt l
fort E (-oo, -1]. The second inequality in (29.72) follows from A±( 'Yt)::::; Area (g (t)).
D
8.2. Backward cylinder limit.
In this subsection we prove the following.LEMMA 29.24 (Existence of an asymptotic cylinder). For any ti --+ -oo, any
minimizer 'Yt, of I ( · ; g(ti)), and any Pi E 'Yt,, the sequence of solutions ( 52 ,
g(t + ti),Pi) subconverges in the Cheeger-Grnmov sense to a static fiat cylinder
(JR x 51 (r 00 ), g 00 (t), p 00 ), t E (-oo, oo), where r 00 is the radius of the circle. More-
over, under this convergence, 'Yt, converges to an embedded geodesic loop "( 00 , which
we may assume to be { 0} x 51 ( r 00 ). Hence, for any c > 0 there exists TE < 0 such
that for each t::::; TE there exists an €-neck in (5^2 ,g(t)) centered at any Pt E 'Yt in
the sense of Definition 18.26 of Part III.Let ti--+ -oo, let the loop 'Yt, be a minimizer ofl( ·; g(ti)), and choose a point
Pi E 'Yt.- Recall from (29.69) that inj(5^2 ,g(t)) ~ Jc fort E (-oo, -1]. From this
and the boundedness and positivity of the curvature, we may apply Hamilton's com-
pactness theorem to conclude that the sequence of solutions (5^2 , g(t+ti),pi) subcon-
verges in the Cheeger- Gromov sense to a complete eternal solution (N~, g 00 (t), p 00 )
with bounded nonnegative curvature. That is, after passing to a subsequence, there
exists an exhaustion Ui of N 00 by open sets and diffeomorphisms
(29. 73) <fJi : ui --+ 52such that <p:g(t +ti) converges to g 00 (t) in each Ck on compact subsets of N 00 x
(-oo, oo). Now, by (29.60), (29.66), and (29.72), we have that the geodesic curva-
tures satisfy
(29.74) .lim /'\,('Yt;) = 0.
i-tooFrom this and from (29.66) again, we have that 'Yt, converges to a geodesic loop "( 00 ;
that is, i.pi^1 ("ft;) converges to 'Yoo in C^00 • Since 'Yoo is the limit of embedded lo ops,
'Yoo has at most self-tangencies and no transverse intersections. Then Lemma A.1.2
in [140] implies that 'Yoo is an embedded geodesic loop.
By (29.72) we have that 'Yt divides 52 into differentiable disks 5i(t) and 5:(t)
with areas satisfying A±bt) ~ cltl, where c > 0. Thus there exist points qt E
5i(ti) and qi E 5: (ti) such that limi-too dg(t;) (qt, Pi ) = oo and limi-too dg(t;) (qi, Pi)
= oo. Let ei = d 9 (t,) (qt, qi) and let /3i : [O, ei] --+ 52 be a unit speed minimal geo-
desic from q: to q;. Since /3i n 'Yt, i:- 0) we may choose a point Oi E /3i n 'Yt,.
Then there exists si E (O,£i) such that oi = /3i(si)· By passing to a subse-
quence, we may assume under t he Cheeger- Gromov convergence of (5^2 ,g(ti),pi)