1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

Lecture 3. The Weinstein Conjecture in the Overtwisted Case

Up to this point we described some of the results which we would like to obtain and
developed some basic analytical machinery. Since there are many ingredients for
the proofs it makes sense to give a complete picture of the different steps involved.
We are given a closed three-manifold and an overtwisted contact form,. Denote
the associated contact structure by ~ and the Reeb vector field by X. The strategy
of the proof is now based on a certain number of non-obvious ingredients and ideas:

l. Existence of small Bishop disks near elliptic complex tangencies on surfaces

in almost complex manifolds.

2. The filling property of families of prolonged Bishop disks.


3. The apriori energy control of Bishop disks.


4. The apriori non-existence of large Bishop disks.


5. A bubbling-off analysis.
   As we shall see all these ingredients boil down to the fact that the analysis does
   not work and therefore, as a consequence of a bubbling-off analysis, there has to be
   a periodic orbit.
   Now we have to elaborate on the above points.

An Explicit Local Bishop Family

By assumption there exists an embedded disk D, the overtwisted disk, such that

T8D c (l8D

and

T z'D c/.. ( z for Z E 8D.

The first crucial observation is the following. It is corollary of the Giroux elimination

lemma, see [35, 19, l].

Proposition 3.1. Given an overtwisted contact form ,\ with associated contact

structure there exists an embedded disk D with the following properties.

• There exists a unique point e E D with T e D = ( e and moreover

T8D c (18D

and

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