- THE QUANTITY Q MUST BE IDENTICALLY ZERO 127
by (29.7), we have a2w as2 > O; i.e., s H W(s- , t) is convex. On the other hand, from
(29.5), we have thatIn v ( s, e , t) = 2 In cosh s + In v ( ~ ( s) , e, t) ,
where ~ is the latitude on S^2. Thus
8~v a~v
----a;-(s, e, t) = 2tanhs + sechs~(~, e, t)since~~= sechs. Since 18 J~vl(t):::; IV'lnvl 952 (t):::; Cv^1!^2 < C for each t E(-oo,-1] by (29.34), this implies that
(29.205)
aw
lim a:v (s, t) = -47r and
s-+-oo us
lim ~(s, t) = 47r.
s-+oo usTherefore a;sif > 0 implies that the derivative of the circular averages satisfies
(29.206) 1
8
8: (s,t)I < 47r fort E (-oo,-1].
In cylindrical coordinates, definevi(s, e, t) ~ Ki-^2 v(s +In Ki, e, t +ti),
so that v;^1 (t)gcyl = v;^1 (t)geuc is the sequence of solutions in (29.119) and let
v 00 (t) = limi-+ 00 vi(t). Formula (29.118) implies that v 00 (s, e, 0) converges to A
uniformly as s---+ oo. Hence there exist so and io such that vi(s, e, 0) :::; 2A for all
i 2': io, s 2': so, and e. In particular,
v(so+lnKi,e, ti) :::;2AKf,
which tends to zero. (This is related to the fact that, backward in time and in the
cigar region, the conformal factor tends to infinity.) So we have
W(so +In Ki, ti) :::; 27r ln(2AKf),
which tends to -oo.
We shall compare the averages over the circle { s = s 0 + ln Ki} in the cigar
region and over the circle {s = si} passing through m(qi), where mis Mercator
projection. Let
(29.207)Since ri ---+ 0, we have Si ---+ -oo. By the Case 2 hypothesis , Si -In Ki ---+ oo. Hence,
Si > so + In Ki for i sufficiently large.
Claim 1. There exists a constant C < oo such that
(29.208) 27rlnμ < W(si,ti):::; C for all i.
Proof of Claim 1. We have already observed the lower bound. To prove the
upper bound by contradiction, we assume that W(si, ti) ---+ oo.
Since lims-+-oo e^28 v(s, e, t) E (0 , oo), we have lims-+- oo(W(s, t) + 47rs) E JR for
each t. In particular, lims-+-oo W(s, t) = oo. We also have
(29.209) lim W(s,ti) = j 1nv 00 (s,B)de = 27rlnμ
i-+oo 51