Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let $(M,\omega )$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\text{RManBase}$ is a $\text{CR}-$manifold of dimension $2n+1$ and $\omega$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^{\omega }\subset T^{\ast }M$ be the canonical symplectization of $(M,\omega )$ and $M$ be identified with the zero section of $M^{\omega }$. Then $M^{\omega }$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $$\mathbb{C}$\ni t+i\sigma\mapsto \phi_p(t+i\sigma)=\sigma\omega_{g_t(p)}$,where$p\inM$isarbitraryand$g_t$istheflowgeneratedbytheReebvectorfieldassociatedtothecontactform$\omega$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^{\omega }$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^{\omega }$ if all the parametrizations ${\varphi }_p(t+i\sigma )$ defined above are holomorphic on ${\varphi }_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^{\omega }$, then the real function $E$ on $M^{\omega }\subset T^{\ast }M$ defined by the condition $\alpha =E(\alpha ){\omega }_{\pi (\alpha )}$, for each $\alpha\in M^\omega$,isacanonicalequationfor$M$ which satisfies the homogeneous Monge-Amp`ere equation $(d{d}^{c}E{)}^{n+1}=0$. We also prove that if $M$ and $\omega$ are real analytic then the symplectization $M^{\omega }$ admits an unique maximal adapted complex tube.