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 ${HT(M) \subset TM}$. Let ${M^\theta \subset T^* M}$ be the canonical symplectization of (M, θ) and let M be identified with the zero section of M ^θ. Then M ^θ is a manifold of real dimension 2(n + 1) which admits a canonical foliation by surfaces parametrized by ${\mathbb{C} \ni t+i\sigma\mapsto \phi^{\theta}_{p}(t+i\sigma)=\sigma\theta_{g_t(p)}}$, where ${p \in M}$ is arbitrary and g _t 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 M ^θ. We say that the pair (U, J) is an adapted complex tube on M ^θ if all the parametrizations ${\phi^{\theta}_{p}(t+i\sigma)}$ defined above are holomorphic on ${(\phi^{\theta}_{p})^{-1}(U)}$. In this paper we prove that if (U, J) is an adapted complex tube on M ^θ, then the real function E on ${M^\theta\subset T^*M}$ defined by the condition ${\alpha=E (\alpha)\theta_{\pi(\alpha)}}$, for each ${\alpha \in M^\theta}$, is a canonical defining function for M which satisfies the homogeneous Monge–Ampère equation (dd^c E)ⁿ⁺¹ = 0. We also prove that if M and θ are real analytic then the symplectization M ^θ admits an unique maximal adapted complex tube.