Analytic discs attached to conormal bundles of CR-manifolds
For a pseudoconvex hypersurface \(M\) in \({\mathbb{C}}^n\), small analytic discs attached to \(M\) are in fact contained in \(M\) [see also \textit{K. Diederich} and \textit{J. E. Fornaess}, Invent. Math. 39, 129-141 (1977; Zbl 0353.32025)]. The author generalizes this result, namely: If \(M\) is a CR-submanifold of \( X= {\mathbb{C}}^n\), if \(z_0\in M\), if we consider a small analytic disc \(A\) (\(C^1\) up to the boundary ) attached to \(M\) through \( z_{0} \), if \(p_{0}\) is a point in the conormal bundle to \(M\) in \(X\) (minus the zero element) (=: \(T\)) whose image through the canonical projection is \( z_{0} \), if there exists a lift \(A^{\star}\) of \(A\) attached to \(T\) through \( p_{0} \) and if the Levi-form of \(M\) has a constant number of negative eigenvalues in a neighbourhood of \(p_{0}\), then, in fact, \(A \sqsubset M\) and \(A^{\star}\) is contained in \(T\).
scientific article; zbMATH DE number 1044220