边界CR叶理的传播与附加解析圆流形的Morera型定理
Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs
摘要：
我们证明了通用的同调非平凡的$(2n-1)$ -参数家庭的分析光盘附加其边界的CR流形$Ω$在$\mathbb C^n, n \le 2$测试CR功能：如果一个光滑函数$Ω$扩展解析内每个分析光盘，然后它满足切向CR方程。特别是，我们回答，在真正的解析范畴，两个开放的问题：在平面域的解析函数的特征（带问题），并在$\mathbb C^n$（一个猜想Globevnik和Stout）域的全纯函数的边界值的特征。我们还刻画了复曲线在$\mathbb C^2$作为真正的2流形admiting同调非平凡的1参数家庭的附加解析盘。的证据是基于减少到一个问题的传播退化的CR叶状环面状流形。
英文摘要：
We prove that generic homologically nontrivial $(2n-1)$-parameter family of analytic discs attached by their boundaries to a CR manifold $Ω$ in $\mathbb C^n, n \le 2$ tests CR functions: if a smooth function on $Ω$ extends analytically inside each analytic disc then it satisfies the tangential CR equations. In particular, we answer, in real analytic category, two open questions: on characterization of analytic functions in planar domains (the strip-problem), and on characterization of boundary values of holomorphic functions in domains in $\mathbb C^n$ (a conjecture of Globevnik and Stout). We also characterize complex curves in $\mathbb C^2$ as real 2-manifolds admitiing homologically nontrivial 1-parameter families of attached analytic discs. The proofs are based on reduction to a problem of propagation of degeneracy of CR foliations of torus-like manifolds.
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