Moreover,LECTURE 2. ANALYTICAL TOOLSE(vk I D1) 2 ~ E(u)
8svk + l8tvk = 0 on D1 2
JD, vk_d>.--tO.
261Now applying Theorem 2.4 we obtain a (non-trivial) finite energy plane w: C -+
IR x M by a C~c-converging subsequence of the sequence (vk)· It follows
Ws + Jwt = 0 on C
o < E(w) ~ E(u) < oo.
But we also can deduce
r w*d).. ~ lim r vk,d>. = 0.
}BR k->oo}D~This implies that w(z) = x(f(z)) for some orbit x of :i; = X(x) and some function
f: C-+ IR. Since
da = (w* >.) o i = df o i,
we deduce that a+ if: C -+ C is holomorphic. The map cannot be constant since
otherwise w would be constant implying E(w) = 0.
We compute for cp E I:
[ w*d(cp>.) = [ cp'(a)da /\ dj.It is now an easy exercise that for a non-constant holomorphic map a+ if and
cp E I: with cp' > 0 we have
[ cp'(a)da /\ df = oo.This contradiction gives uniform gradient bounds. Consequently we have uniform
C^00 -bounds on [1,oo) x S^1.
Define
Then
vk: [-ski oo) x S^1 -+ IR x M
vk(O, 0) E {O} x M8svk + ]8tvk = 0
E(vk) = E(u) < oo
r Vk(O,·)*>.--->m
Js,r vk,d>. --t 0.
1[-R,R]xSBy the C^00 -Ascoli-Arzela we find taking a suitable subsequence without loss of
generality
in C 1 ~c(IR x S^1 , IR x M)