98 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONS
that for any c > 0 there exists Tc :::::; - 1 such that g(t) 2". C^1 g 52 fort:::::; Tc. Hence
c 2". L 9 ct,)(s- _) 2". c-^1 l^2 L 9 2 (s- _) 2". l.9997rc-^112 ,
ti,Tti S ti,Tti
which is a contradiction. D
Proof of th e claim. Since Rg(t) 2". 0, from (29.90) we obtain that the function
r H L^9 <•>~
3
~rl is nonincreasing. Now choose the basepoint Pt in the statement of
Lemma 29. 24 to b e Pt E /t, so that dg(t)(Pt,xt) =Pt, where Pt is defined by
(29.81) and where Pt 2". cit! holds by (29.85). Note that, by Lemma 29 .26, the
minimizer satisfies
(29.92)
for all t:::::; -1. Without loss of generality, we may assume that the 8B~~) (Pt) ~ a.t
t
are piecewise smooth in addition to being connected; otherwise we replace Pt by
a suitably close value (the reader may check that the argument still goes through
without ch ange). By Lemma 29. 26 we have Lg(t)(a.t):::::; C.
To complete the proof of the proposition, we consider three cases.
(1) If rf: 2". Pt, then (29.90) tells us that
(29.93)
(2) If Pt - 10 < rf: < Pt, then by the geometry of the neck , the length
L 9 (t)(s-t,rt _)is nearly 27r.
(3) If rf: :::::; Pt - 10 , t hen by (29.92), the b all B^9 x, ~) (pt) is contained in 52 -
B^9 ~)(rf:). Hence Area 9 2 B^9 ~)(pt):::::; 27r(l-ct)· This implies r{ 2". p{. Thus,
xt s xt
(1) with+ replacing - implies that L g(t)(s+ t,rt +):::::; C.
This completes the proof of the cl aim and h ence of the proposition. D
The following result sh all b e used in the proof of Proposition 29.36.
PROPOSITION 29 .30 (The plane is not a backward pointwise limit). For any
choice of point p E 52 , it is not possible for g(t) = v[t) g 52 to converge in C^00
on compact sets of 52 - {p} to l7* ( Ageuc) as t -+ - oo, where A > 0 and where
l7 : 52 - {p} -+ IR^2 denotes stereographic projection.
PROOF. Let g(t) = (l7-^1 )*g(t) = -u[t)geuc· We prove the proposition by contra-
diction. Suppose that v-^1 (t) converges to a positive constant A as t-+ -oo. Then,
for each k E N (which we shall assume is sufficiently large), there exists tk :::::; -1
such that
(29.94)
where 0 denotes the origin of IR^2. Let 'Yt be an isop erimetric curve. We may
assume, without loss of generality, that the antipodal point q ~ l7-^1 ( 0) of p satisfies
q E 5i('Yt).
Define tk ~ min{tk, -~7rk^2 }, where c is given by (29.72). We claim that if
t E ( -oo, tk], then
(29.95)
