Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Abstract. Let (M,θ) be a pseudo-Hermitian space of real dimension 2n+1, that is M is a CR-manifold of dimension 2n+1 and θ is a contact form on M giving the Levi distribution HT(M) ⊂ TM. Let Mθ ⊂ T∗M be the canonical symplectization of (M,θ) and let M be identified with the zero section of Mθ. Then Mθ is a manifold of real dimension 2(n + 1) which admits a canonical foliation by surfaces parametrized by C in t + iσ →φθ p(t + iσ) = σθgt(p), where p ∈ M is arbitrary and gt is the flow generated by the Reeb vector field associated to the contact form θ. Let J be an (integrable) complex structure defined in a neighbourhood U of M in Mθ. We say that the pair (U, J) is an adapted complex tube on M if all the parametrizations φθ p(t + iσ) defined above are holomorphic on (φθ p)−1(U). In this paper we prove that if (U, J) is an adapted complex tube on Mθ, then the real function E on Mθ ⊂ T∗M defined by the condition α = E(α)θπ(α), for each α ∈ Mθ, is a canonical defining function for M which satisfies the homogeneous Monge–Amp`ere equation (ddcE)n+1 = 0. We also prove that if M and θ are real analytic then the symplectization Mθ admits an unique maximal adapted complex tube.