Leafwise de Rham cohomology of generic Reeb foliations

In this paper, we prove that there exists a residual subset of contact forms $\lambda$ (if any) on any compact connected orientable manifold $M$ for which the foliation de Rham cohomology of the associated Reeb foliation has $H^0(F_\lambda) \cong \mathbb R$. We also prove the same triviality for a generic choice of contact forms with fixed contact structure $\xi$. This vanishing result of $H^0(F_\lambda)$ is also equivalent to the statement that the Lie algebra of the group of strict contactomorphisms is isomorphic to the span of Reeb vector fields, and so isomorphic to the 1 dimensional abelian Lie algebra $\mathbb R$. On the other hand, we derive the rank of $H^1( F_\lambda)$ is infinite whenever $\lambda$ admits a closed Reeb oribt, i.e., whenever Weinstein's conjecture holds. In particular we prove that $H^1(F_\lambda)$ is infinite dimensional for all contact form $\lambda$ in dimension 3, thanks to Taubes' proof of 3-dimensional Weinstein's conjecture.