Lecture 4. The Weinstein Conjecture in the Tight Case
4.1. Sketch of the proof
In the case that the contact form is tight and n2(M) "/; 0 one can find as a conse-
quence of the sphere theorem an embedded non contractible two-sphere. Applying
the Giroux elimination process we obtain a new sphere F having two elliptic tan-
gencies with nice normal form in suitable coordinates. In this case we can start a
Bishop family at each of the two singularities. Assume we have uniform gradient
bounds for the two families. If we can show that the Bishop families match up when
they meet we would be able to construct a continuous map [-1, l] x D-> ~ x M,
where { -1} x D is mapped to one singularity and { 1} x D to he other. Collapsing
top and bottom of the cylinder we obtain a map from the closed three-ball D^3 to
~ x M, which induces a homeomorphism S^2 = 8D^3 -> F. This however implies
that that F is contractible, contrary to our assmption. Hence we cannot have gradi-
ent bounds for our Bishop family. Gradient bounds at the boundary can be obtained
as before. Hence we obtain the "exploding" disk family as in the overtwisted case.
ee+Figure 12. The characteristic foliation on the non-contractible sphere F after
applying the Giroux elimination process.87