76 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS
The linearisation T of the Cauchy Riemann operator is given by
T(h, k) + (hs + iht + ak, ks+ ikt +bk).
Here a, b E C^00 (D, .CR(C)). The Cauchy Riemann operator
[)j : B ____, LP(D, C^2 )u __,Us+ J(u)uthas the particular feature that J(z, 0) = i for lzl ::::; 1. This feature was of course
the crucial ingredient for the useful results derived in the previous section. Ellip-
tic regularity theory implies that every solution of the nonlinear equation is smooth.
Now we would like to study the solution set 871 (0) C B near uo using the
implicit function theorem.
For this purpose we work with the chart if.> that we constructed previously.
Recall that if> : Tu 0 B :J B __, B ; <f.>(f) = exp (f) satisfies <f.>(O) = uo and D<f.>(O) =
Idr,, 0 13. Since ker DDJ(uo) is finite dimensional we find a topological complement
X in Tu 0 B.
Fix now an element (ho, ko) E ker DDJ(uo) with ko(z) f= 0 for all z ED. For c > 0
sufficiently small we define the smooth map
H: (-c, c) x X----+ LP(D, C^2 )
( r, f) I---) DJ( if>( r(ho, ko) + !)).
We observe that H(O, 0) = DJ(u 0 ) = 0 and
DH(O,O)(s,g) DDJ(uo) D<f.>(O) · [s(ho, ko) + g]
= s DDJ(uo)(ho, ko) + DDJ(uo) g
= DDJ(uo) g.
Since D2H(O,O) : X __, £P(D,C^2 ) ; g I---) DDJ(u 0 )g is an isomorphism we find
by the implicit function theorem a neighbourhood X' of zero in X , i:;' ::::; c and a
unique smooth map
a: (-i:;', i:;') ----+ X'
so that
H(r, a(r)) = 0 and a(O) = 0.
Define
u: (-c',i:;') x D ____, C^2
(r, z) I---) <f.>(r(ho, ko) + a(r))(z ).
We note that TI---) u(r, .) is a smooth path in B with u(O,z) = u 0 (z) = (z,O) and
a JU( T,. ) = 0. By elliptic regularity the maps z I---) u( T, z) are smooth.
Let us calculate the derivative Du(O, z):