伪厄米流形辛化的适应复管
Adapted complex tubes on the symplectization of pseudo-Hermitian manifolds
摘要：
设$(M,ω)$是实维$2n+1$的伪厄米空间，即$\RManBase$是$2n+1$的$\CR-$流形，$ω$是$M$上的一个切触形式，给出了Levi分布$HT(M)\subset TM$ .设$M^ω\subset T^*M$是$(M,ω)$的规范辛化，$M$与$M^ω$的零截面相同.则$M^ω$是实维$2(n+1)$的流形，其允许由$\mathbb{C}\ni t+iσ\mapsto ϕ_p(t+iσ)=σω_{g_t(p)}$参数化的表面的正则叶理，其中$p\inM$是任意的，$g_t$是由与接触形式$ω$相关联的Reeb向量场生成的流。设$J$是定义在$M^ω$中$M$的邻域$U$中的（可积）复结构。我们说对$(U,J)$是$M^ω$上的{适应复管}，如果上面定义的所有参数化$ϕ_p(t+iσ)$在$ϕ_p^{-1}(U)$上是全纯的。本文证明了如果$(U,J)$是$M^ω$上的一个自适应复管，则对于每个$α\in M^ω$，由条件$α=E(α)ω_{π(α)}$定义的$E$在$M^ω\subset T^*M$上是满足齐次Monge-Ampère方程$(dd^c E)^{n+1}=0$的$M$的正则方程.我们还证明了，如果$M$和$ω$是实解析的，则辛化$M^ω$允许唯一的最大适应复管。
英文摘要：
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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