70 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICSJ(x,y,z) ( Jx ) - ( ~1 )
and we continue J arbitrarily onto the whole manifold. In the above coordinates
the nonlinear Cauchy Riemann equation looks as follows:
(10)
(11)
Osa - OtZ - XOtY 0
08 Z +Ota - XOtX
08 X - OtYOsY + OtX
The required boundary condition is00
0
0.fore E [O, 27r]. We leave it as an exercise for the reader to verify that the maps
UT : D -----+ IR^4j T > 0 , s^2 + t^2 :::::; 1
are solutions of the boundary value problem (10) and (11). We also have uT(D) n
up(D) = 0 whenever p -=J. T.
Implanting the above construction into the manifold the next point we have to
study is the question of how far the Bishop family can be continued. An important
ingredient is the implicit function theorem.The Implicit Function Theorem near an Embedded Disk
Let us start with a quite general situation. Assume that (W, J) is an almost complex
manifold of real dimension 4. Suppose F C W is a 2-dimensional totally real surface
without boundary, say, and w : D -+ W an embedded J-holomorphic map withw(8D) c F.
The aim is to desribe the nearby holomorphic maps. In order to achieve that
it is useful to derive a suitable normal form.
Theorem 3.2. Let w: D-+ W be a solution of(12) W 8 + J(w)wt = 0 on D and w(8D) CF,
which is an embedding. Then there exists an open neighbourhood U of w(D) c W
and a diffeomorphism <p : U -+ V C C^2 onto an open neighbourhood of 0 E C^2 so
that(<po w)(z) = (z, 0) for all z ED.
Moreover if u: D-+ W is another solution of (12) with u(D) c U then v :=<po u
satisfies