88 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSOf course the argument only works if we prove some uniqueness result, which
would imply the matching of the Bishop families if they meet.
4.2. Global uniqueness for families of pseudoholomorphic disks
Starting with an embedded solution of the Cauchy Riemann equation, with local
normal form having k = 0 we have seen that all closeby solutions are given by the
implicit function theorem. Moreover two closeby holomorphic disks are either iden-
tical or disjoint. We refer to this situation as local existence and local uniqueness
for disk families.
In this section we will study the following more global question: Assume u 0 is
as just described and v is another embedded solution of the boundary value prob-
lem, which is not necessarily closeby such that the associated disk Dv intersectsDo = Duo. We shall show that under some geometric assumption on F , namely
that F is a subset of a pseudoconvex three-dimensional sub manifold M of W, thedisk Dv will necessarily intersect all Dr for T E (-i:;, 0), for some small c > 0 un-
less Dv = D 0 , provided the disks lay on the same side of M in W. This has the
following striking consequence:If (D;.)rE(-c:,c:) and (D;)rE(-c:,c:) are two disk families associated to the same
ellipt,ic boundary value problem such that D~ro n D;. 0 = 0 for all To E (-i:;, 0) and
D6 n D6 =:j:. 0, then D6 = D6. Hence, perhaps changing the parametrisation we can
glue this two disk families to obtain a possibly larger disk family.Assume (W, J) is an almost complex four-manifold and F c W is .a totally real
submanifold. We assume that u 0 : D __, W is an embedded holomorphic disk with
boundary in F, i.e.
81uo = 0 on iJ and uo(8D) c F.
Suppose also that
k = 0.
Denote by (Dr)rE(-c:,c:) the local disk family associated to u 0 , with Do = u 0 (D).
Let v : D ·---+ W be another solution of the boundary value problem. We will not
assume that the associated disk Dv is close to our disk family. Instead we assume
e·' ~~.::-~~mtw..:?:"W..S.~*;,,,_~$:'-::-::...;::e:....,_ __ ,
·••'0#<\,'01 "~~~~'*®"''·"'
················································
···························-·······························Figure 13. A schematic picture of the Bishop family.