1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. THE WEINSTEIN CONJECTURE IN THE OVERTWISTED CASE 79

3.3. A short summary and the prolongation of the Bishop family

Let us sum up what we know up to this point. We are given a closed three-manifold
M equipped with a contact form .A, so that its underlying contact structure ~ is
overtwisted. Starting with an overtwisted disk we modify, as dicussed in length,
so that we obtain a new disk, called D C M, containing only one singular point
e, which is elliptic. For a proper orientation of the characteristic foliation it is
a source, having the boundary as a limit cycle. Moreover in suitable coordinates
transporting e to 0 E IR^3 the disk D near e looks like the graph of the function
(x,y) -+ -~xy. Recall that the relevant contact form on IR^3 is dz + xdy. Near
0 we take as complex multiplication the one mapping over the point (x , y , z) the

vector (1, 0, 0) to (0, 1, -x). The associated IR-invariant almost complex structure

is then integrable and an explicit Bishop family can be constructed. Moreover, the
linearized Cauchy-Riemann operator at such a solution can be computed to have

index 4. It is important to note that the linearized operator has an upper triangular

form

[ ~l ~]

where T 1 and T 2 are surjective Fredholm operators of index 3 and 1 respectively.
Here T 1 corresponds to the direction tangential to a given Bishop disk and T2 normal

to a Bishop disk. The "3" in 4 = 3 + 1 then corresponds to the three-dimensional

reparametrisation group of the holomorphic disk. Consequently the Bishop family
represents the only solutions near the singularity (modulo) reparametrisation. This
Bishop family is then implanted by the chart into M and J is extended as an
admissable complex multiplication to the whole of M.
The remaining part of the proof now consists in prolonging the Bishop family
as far as possible.
The problem we are studying is

(14) u: D -+IR x M

us + l(u)ut = O

al(8D) = 0

u : 8D -+ 8D \ { e} has winding number 1.

In addition we require that u is homotopic with boundary conditions as just de-
scribed to disks in the Bishop family. In particular this implies that for these disks

k = 0. Take a leaf of the characteristic foliation other than the boundary, say £.

Since e is nicely elliptic we parametrize £ by arc length (starting from e) taking
the metric d.A o (Id x J) +.A 0 .A on M. We then write f(T) for points on£, where
T is the distance to e along £. Clearly the length of£ is oo. We parametrize the
Bishop family in such a way that FT contains £( T) for T ;:::: 0 sufficiently small. Here
FT is an unparametrized Bishop disk. The question is now how far can the Bishop
family be prolonged. Invoking the implicit function theorem we find a number
To E (0, oo) U { oo }, so that there exists a smooth family of solutions Fn T E [O, To)
having the following properties:

(15) f(T) EFT

FT is an embedding for T(O, To).

Moreover every FT for T > 0 is of the form u(D), where u solves (14) and is an

embedding. The neighbouring disks are then given by a one-parameter family of