68 H. HOFER, HOLOMORPHIC CURVES AND DYNAMICS• The distribution T(V \ { e}) n ~, which is one-dimensional defines a foliation
having e as a sink-source type rest point and 8V as the unique limit cycle.- There exist open neigbourhoods U(O) of 0 in IR^3 and V(e) in M and a dif-
feomorphism 'ljJ : U(O) -t V(e) such that 'ljJ* A = dz+ xdy, 'ljJ(O) = m and
'ljJ-^1 (V) is the graph of the function (x, y) -t -~xy.
The proof of this proposition is not an easy task and we refer to the above listedreferences for more detail. The proof starts with an overtwisted disk. By a c=-
small perturbation keeping the boundary fixed the characteristic foliation becomes
Morse-Smale. Then one looks at connecting orbits between singular points. If
the induced orientations from (~, d.X) on the disk at the singularities are related by
continuation, t hen the two singularities can be removed by a C^0 -small perturbation
in the neighbourhood of the connecting orbit (Giroux elimination).
Again perturbing by keeping the boundary fixed one obtains a Morse-Smale
fl.ow. These two perturbation may create new limit cycles, which of course bound
disks. These smaller disirn are of course overtwisted, but contain fewer singular
points. During this process also a situation may arise where one has to produce
in a controlled way a pair of singular points. Without going further into details it
becomes hopefully clear that the result is not straightforward.
The point e is a non-degenerate elliptic tangency between ~ and V. If we pick
an admissable complex multiplication for ~ and take the associated almost complexstructure Jon !RxM we see that {O} x V has a complex tangency at (0, e) which is of
elliptic type. In the integrable case, i.e. if J is a complex structure, Bishop showed
that a one-dimensional family of small holomorphic disks with their boundary on
{O} x V pop out of the singularity (0, e).
The first step is to get started using the Bishop family coming out of the
singularity e.
By the previous proposition we may assume that the contact manifold is just
JR^3 with the standard contact form .A= dz+ xdy. Moreover the surface Vis given
Figure 6. The Giroux elimination process. The nontrivia l foliation is per-
turbed by a c^0 small perturbation of the surface into a simpler one.