- CLASSIFYING THE BACKWARD POINTWISE LIMIT 105
ThusThis completes the proof of Claim 1.
Now, by Corollary 6.3 in [194], (29.110) implies that^8 t-: and^8 ~f' are boundedharmonic functions on IR^2 and h ence are constant. Thus ln v 00 (s, B) = C 5 s + C 6.
We conclude tha t(29.112) Voo = μ eBs,
where μ > 0 and B are constants.
In view of (29.108), the proposition follows fromClaim 2. B = 0.
Proof of Claim 2. By (29.112) and (29.108) we have thatVoo('l/J, B) = μe^88 Sech^2 S.Since we have C^3 '°' convergence of v(t) to v 00 on compact subsets of 52 - {N, S},
the gradient estimate (29.34) holds for v 00 , so that
C j'Vvool;2 _ cosh2s (8voo)
2
> - --- -- -_ μ e Bs(B - 2 tan h s )2- V 00 V 00 OS
for s E IR.This implies that B = 0 or B = ±2. Suppose that B = ±2. Then v 00 = μ e±^2 s
implies that as t --t -oo we have that )t) g 52 limits on 52 - { N} to
V,_l =t=2s(d^2 dB2)
00 9 cyl = μe S + = μgeuc·This contradiction to Proposition 29.30 yields B = 0. 0
As a consequence of the proposition, we can interpret the gradient estimate
(29.34) as follows.
COROLLARY 29.37. Suppose that (5^2 , g(t)), t E (-oo, 0) , is a nonround solu-tion. Then there exists C < oo such that
(29.113) (:elnvr :::;C on5^2 x(-oo,-1].
PROOF. Since g 52 ('ljJ,B) = d'ljJ^2 + cos^2 '1j;dB^2 , (29.34) says that
(~~r :::;j'Vvj~ 2 cos^2 '1j;:::;Cvcos^2 '1j; on5^2 x(-oo,-l].
On the other hand, since g(t) is nonround, by Proposition 29.36, we have that
μcos^2 'ljJ = V 00 (1/J, B):::; v('l/J, B, t)for someμ> 0. 0