92 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSnear the point zo where Dv and Do intersect for T < 0 close enough to zero. This
is equivalent to
(38) a(z)
b(z)where z' E Dia and z E Di, for some sufficiently small c1 > 0. Of course by
assumption this problem does not have any solution. Since aT is C^00 - close to
'ljJ we see that ar : D"to ___, D is a diffeomorphism onto its image which is some
neighbourhood of z 0 in D. Hence equation (38) can be rewritten as
(39) b(z) = br ( a;^1 ( a(z))).
Here z E Di,. Consider the mapA : [O, l] x Di, -----+ CA(B,z) := b(z) + ()8 - (1-B)br(a;^1 (u(z))).
For T and 8 small enough we have 0 tf_ A([O, l] x 8D"1;_). Let us check this quickly:0 E A({O} x 8D"1;_) for some T < 0
is equivalent to0 = b(z) - br(a;^1 (a(z)))
but we said above that this problem has no solution.For 0 < () :::; 1 and T = 0 we have
A(B, z) = b(z) + () 8
since b 0 = 0, but we saw before that this expression is nonzero if 8 > 0 is sufficiently
small and z E Di,. Then it is also nonzero if 0 < () :::; 1 and T < 0 with I T I small.
Using the homotopy invariance of the degree we obtain therefore
deg(A(O, *), D~, 0) deg(A(l, *), D"1;_, 0)deg(b + 8, D~, 0)
deg(b, D~, -8)
ko
2
By the solution property of the degree we deduce that Dv intersects Dr giving a
contradiction. Hence we have proved the following:
Proposition 4.1. Let J be an almost complex structure on C^2 with J(z, 0) = i for
I z I :'.S: 1 and let
(Dr)rE(-e,e) = (u(T,D))rE(-o,o)be the family of holomorphic disks with u(T,8D) C [)D x JR. near u(O,z) = (z,O)
given by the implicit function theorem.
Assume that A C D is an open and connected subset and the embedding
v: A-----+ C^2
satisfies