Regularity of discs attached to a submanifold of \(\mathbb{C}^n\)
Let \(0<\alpha<1\) and \(\{M_\xi\}_{\xi\in \partial D}\) be a family of generating \(CR\)-submanifolds of \textbf{C}\(^{m+n}\) of \(CR\)-dimension \(m\) and let \(p:\partial D\to\) \textbf{C}\(^{m+n}\) be a \(C^\alpha\) map such that \(p(\xi)\in M_\xi\) for every \(\xi\in \partial D\). The author proves the following results: Theorem 1. A \(C^\alpha (\alpha>{1\over 2})\) closed curve \(p(\partial D)\) in a \(C^\alpha\) family of \(C^2\) generating \(CR\)-submanifolds \(\{M_\xi\}_{\xi\in \partial D}\) of \textbf{C}\(^{m+n}\) is regular if and only if there exists a \(C^\alpha\) fiberwise holomorphic change of coordinates of \textbf{C}\(^{m+n}\) such that in the new coordinate system all partial indices of the closed curve \(p(\partial D)\) in \(\{M_\xi\}_{\xi\in \partial D}\) exist and are all greater than or equal to \(-1\). Theorem 2. A \(C^\alpha (\alpha>{1\over 2})\) closed curve \(p(\partial D)\) in a \(C^\alpha\) family of \(C^2\) generating \(CR\)-submanifolds \(\{M_\xi\}_{\xi\in \partial D}\) of \textbf{C}\(^{m+n}\) is completely regular if and only if there exists a \(C^\alpha\) fiberwise holomorphic change of coordinates of \textbf{C}\(^{m+n}\) such that in the new coordinate system all partial indices of the closed curve \(p(\partial D)\) in \(\{M_\xi\}_{\xi\in \partial D}\) exist and are all greater than or equal to 0. Several applications are also given.
scientific article; zbMATH DE number 1146581
scientific article