My paper with Jo Nelson on cylindrical contact homology for dynamically convex contact forms in three dimensions is now posted here. This paper shows that you can define the differential by counting -holomorphic cylinders for a generic almost complex structure , without any abstract perturbation of the Cauchy-Riemann equation. (We make one additional technical assumption, which can probably be dropped with some more work on the asymptotics of holomorphic curves, and which automatically holds when of the three-manifold has no torsion.) We also prove that , filling a gap in an earlier published proof of this, using the methods in this blog post.
The proof of invariance (i.e. the proof that cylindrical contact homology is an invariant of three-manifolds with contact structures that admit a dynamically convex contact form) is postponed to a subsequent paper. When there are no contractible Reeb orbits, the proof of invariance involves adapting the methods of Bourgeois-Oancea as explained in this blog post. In the presence of contractible Reeb orbits, there is an additional wrinkle; one has to count nontrivial contributions from holomorphic buildings consisting of an index 0 pair of pants branched covered over a trivial cylinder together with an index 2 plane. For an introduction to this, see the second half of this video.