Local holomorphic foliations and partial analyticity of \(C^\infty\) CR mappings.
Consider a smooth CR mapping \(f\) between a generic real analytic submanifold \(M \subset \mathbb{C}^{n}, n>1\), and a real analytic subset \(M' \subset \mathbb{C}^{n'}\). Let \(p \in M, p'=f(p)\), and \(\tau (f)\) be the germ at \((p,p')\) of a smaller complex analytic subset of \(\mathbb{C}^{n+n'}\) containing a graph of \(f\) in a neighborhood of \((p,p')\). Partial analytic degree of \(f\) at \(p\), denoted by \(\deg_{p}f\), is defined to be the integer \(\dim T_{p}(f)-n\), and \(M'\) is said to be \((r,s)\)-flat at \(p\), if it contains a real analytic submanifold of dimension \(r\) passing the point \(p'\) and biholomorphic to the Cartesian product \(N \times D\), where \(N \subset \mathbb{C}^{\nu}\), \(\nu \geq 0\), is a real analytic submanifold and \(D \subset \mathbb{C}^{s}\) is a domain. Here the author establishes an upper estimate of the partial analyticity of \(f\) in terms of the maximal dimension of the local holomorphic foliations contained in \(M'\), which is the main theorem of this paper: Let \(f: M \to M'\) be a smooth CR mapping between a submanifold \(M \subset \mathbb{C}^{n}\), which is real analytic and minimal at \(p \in M\), and a real analytic subset \(M' \subset \mathbb{C}^{n'}\). If \(\deg_{z}f\) is a constant equal to \(s\) in a neighborhood of \(p\), then \(M'\) is \((r,s)\)-flat at \(f(q)\), where \(r \geq \operatorname {rank} \max_{p}f\), for each point \(q\) of an open dense subset of \(M\) near to \(p\).
scientific article; zbMATH DE number 1837445