Contact Geometry of One-Dimensional Complex Foliations

Let V be a real hypersurface of class Ck, k ≥ 3, in a complex manifold M of complex dimension n+1 and HT(V) be the holomorphic tangent bundle to V giving the induced CR structure on V. Let θ be a contact form for (V,HT(V)) and ξ0 be the Reeb vector field determined by θ, and assume that ξ0 is of class Ck. In this paper we prove the following theorem (cf. Theorem 4.2): if the integral curves of ξ0 are real analytic, then there exist an open neighbourhood M0 ⊂ M of V and a solution u ∈ Ck(M0) of the complex Monge-Amp`ere equation (ddcu)n+1 = 0 on M0 which is a defining equation for V. Moreover, the Monge-Amp`ere foliation associated to u induces on V that one associated to the Reeb vector field. The converse is also true. The result is obtained solving a Cauchy problem for infinitesimal symmetries of CR distributions of codimension one which is of independent interest (cf. Theorem 3.2 below).