90 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
We have Tv(z 0 )i = iTv(zo) since J( v(zo)) = J(zo, 0) = i. Now
Tv(zo)(Tz 0 8 D) T v(zo) ·!Rizo= izoIR x {O}
C T(zo,o)Fsince by our assumption D v n D r = 0 for TE (-s, 0). Hence
Tv(zo) · !Rzo = -iTv(zo) · i!Rzo
= !Rzo x {O}.
This implies lm(Tv(zo)) = lm(Tu 0 (zo)) which precisely meanslm(Tv(zo)) = C x {O}.
Therefore we may represent the set v(B) as a graph over a subset of CCC x {O}
for a suitable open neighbourhood B C D of z 0. If B is small enough we deduce
from the first assumption that its image v(B) either projects onto a subset of Dunder pr 1 or onto a subset of C \ fJ. Here comes the second crucial assumption:
(35) We assume that v(B) projects into D.We write v(z) = (a(z), b(z)) for z EB and note that la(z)I ::; 1 for z EB (because
of (35)) and Db(zo) = 0 since lm(Tv(zo)) = C x {O}. We compute over B0 (as, bs) + J(a, b)(at, bt)
(36) = (as,bs)+ (J(a,0)+ fo
1
D2J(a,Tb)bdT) (at,bt)= (as+ iat + o:b, bs + ibt + (Jb).Here (o:x, (Jx) = u; Dd(a, Tb)x dT)(at, bt) · Next we take the projection of equa-
tion (36) onto the second coordinate and deduce(37) 0 = bs + ibt + (Jb where b(8B) CR
Using a version of the the similarity principle, [59, 40], for the case of boundary
values, see [1] we deduce that b = 0 if the oo-jet of b vanishes at zo. Here we assume
without loss of generality that B is connected. Hencev(z) = (a(z), 0) = (uo o a)(z)
where a is holomorphic. Since both u 0 and v are embeddings they differ by a
biholomorphic map of the disk.Assume next that the oo-jet of b does not vanish at zo. If we replace B by a
smaller neighbourhood B' of z 0 in D and we apply a biholomorphic map'ljJ: (D+,(-1,1),0)---> (B',8B',zo),
then we may assume that b is defined on the upper half disk D +. Moreover we mayassume that b(O) = 0, Db(O) = 0 and b((-1, 1)) c R Because of the Similarity
principle we find a nonzero holomorphic map u : Dta ~ C on some smaller halfdisk Dta and a map E n2<p<ooW^1 ,P(Dta,GL(C)) with ((-s 0 ,s 0 )) C GL(IR)
such that