82 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
In addition, there is a sequence zk E D+, Zk ---+ 0 and ck ---+ 0, ck > 0 satisfying
IY'uk(zk) !ck ___. ooIY'uk(z) I::; CIY'uk(zk)I for z ED+ with lz - zkl <ck·
(21)
Here C is a positive constant replacing the constant 2 in (17) due to the composition
with cp. Moreover, in view of (18)
(22)where p 2': 0 is a non negative number replacing the previous pin (18). Abbreviating
Rk = l'i7uk(zk)I we carry out a rescaling argument and define Vk on Dt-IRe(zk)I
by
(23)Then, if Zk = iim(zk)Rk, we have
(24)
Moreover,
(25)IY'vk(zk)I = 1
zk ---+ ip.Since lzkl ---+ p we conclude that 0 E Dc:kRk (zk) n H+ if k is sufficiently large.
Since Vk has uniform gradient estimates and satisfies the Cauchy-Riemann type
equation (20), we conclude uniform estimates for all derivatives of Vk, see [l] and
[40]. Therefore, by the Arzela-Ascoli theorem, possibly after taking a subsequence,
the sequence Vk converges in C 1 ~c to a map v: H+ ---+ JR x M satisfying
(26)
V 8 + ](v)iit = 0 on int(H+)
v(JR) c F
l'i7v(ip)I = 1 and l'i7v(z) I ::; Con H +
E(v) < oo.
We claim that the measure of parameter values of characteristic leaves intersected
by v(JR) c V is equal to l. This then finishes the proof in the case that p < oo.
Indeed, given 8 > 0 we then find an R > 0 such that v[-R, +R] intersects leaves
of total measure at least 1 - 8. Since VJ ---+ v in CJ;'c, vJ[-R, +R] intersects leaves
of measure at least 1 - 28 if j is sufficiently large. Since vk[-R, +R] = uk(Sk),
where Sk = [Re(zk) - ~,Re(zk) +~],we conclude that uk(Sk) intersects leaves
of measure at least 1 - 28. Define now /k = IRe(zk)I + Jik, then /k ---+ 0 and
Uk [-/k, /k] intersects leaves whose associated parameter set has measure at least
1 - 28. We have arrived at the required contradiction to (19) and it remains to
prove the claim.
We take a map cp which maps D \ { 1} biholomorphically onto H+. The com-
position w = v o cp is a smooth map w: D \ {l} ---+ JR x M satisfying (26) with
w(8D \ {l}) c V \ {e }, so that it stays away from e, and E(w) < oo. It has been
shown in [40] that the image of w is compact in JR x M. Therefore, a boundary
value version of Gromov's removable ~ingularity theorem applies, (see [36] and [l])
and we conclude