Contact geometry of one dimensional holomorphic foliations

Let V be a real hypersurface of class C^k, k>=3, in a complex manifold M of complex dimension n+1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let \theta be a contact form for (V,HT(V)), \xi_0 the Reeb vector field determined by \theta and assume that \xi_0 is of class C^k. In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of \xi_0 are real analytic then there exist an open neighbourhood N\subset M of V and a solution u\in C^k(N) of the complex Monge-Amp\`ere equation (dd^c u)^(n+1)=0 on N 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 3.1).