Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds

Let $(M,ω)$ be a pseudo-Hermitian space of real dimension $2n+1$, that is $\RManBase$ is a $\CR-$manifold of dimension $2n+1$ and $ω$ is a contact form on $M$ giving the Levi distribution $HT(M)\subset TM$. Let $M^ω\subset T^*M$ be the canonical symplectization of $(M,ω)$ and $M$ be identified with the zero section of $M^ω$. Then $M^ω$ is a manifold of real dimension $2(n+1)$ which admit a canonical foliation by surfaces parametrized by $\mathbb{C}\ni t+iσ\mapsto ϕ_p(t+iσ)=σω_{g_t(p)}$, where $p\inM$ is arbitrary and $g_t$ is the flow generated by the Reeb vector field associated to the contact form $ω$. Let $J$ be an (integrable) complex structure defined in a neighbourhood $U$ of $M$ in $M^ω$. We say that the pair $(U,J)$ is an {adapted complex tube} on $M^ω$ if all the parametrizations $ϕ_p(t+iσ)$ defined above are holomorphic on $ϕ_p^{-1}(U)$. In this paper we prove that if $(U,J)$ is an adapted complex tube on $M^ω$, then the real function $E$ on $M^ω\subset T^*M$ defined by the condition $α=E(α)ω_{π(α)}$, for each $α\in M^ω$, is a canonical equation for $M$ which satisfies the homogeneous Monge-Ampère equation $(dd^c E)^{n+1}=0$. We also prove that if $M$ and $ω$ are real analytic then the symplectization $M^ω$ admits an unique maximal adapted complex tube.

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