The Hartogs phenomenon on weakly pseudoconcave hypersurfaces
The author investigates the Hartogs phenomenon on weakly pseudoconcave CR manifolds \(M\) of hypersurface type, extending the results, originally obtained by \textit{G. Henkin} and \textit{V. Michel} [J. Math. Pures Appl., IX. Sér. 81, No.12, 1313--1395 (2002; Zbl 1031.32025)] for small domains on a real analytic pseuconcave CR manifold. Here hypersurfaces of class \(\mathcal{C}^2\) are considered and the smallness is substituted by the condition of having only one topological end. Different sheaves of CR functions and distributions on \(M\) are considered, namey the restriction \(\mathcal{O}\) to \(M\) of the sheaf of germs of holomorphic functions on an ambient complex manifold \(X\) and those consisting of germs of functions or distributions on \(M\) which are solutions of the homogeneous tangential Cauchy-Riemann equations. The Hartogs phenomenon for one of these sheaves \(\mathcal{S}\) consist in the fact that, for \(K\) compact in an open subset \(U\) of \(M\), the restriction map \(\mathcal{S}(U)\to\mathcal{S}(U\setminus{K})\) is an isomorpism. This is proved true for \(\mathcal{O}\) when \(M\) is a leaf of a \(\mathcal{C}^2\)-smooth foliation of a Stein manifold \(X\) by weakly pseudoconcave hypersurfaces. This applies in particular to weakly pseudoconcave hypersurfaces that can be described as graphs. In this case, the Hartogs phenomenon is shown to be valid also for continuous CR functions, or CR distribution in case \(M\) is smooth, but in the weaker form that, given \(X\Subset{U}\), there is some \(L\) with \(K\subseteq{L}\Subset{U}\) such that the restriction map \(\mathcal{S}(U) \to\mathcal{S}(U\setminus{L})\) is onto. Finally, also Levi-flat properly embedded \(3\)-dimensional CR submanifolds \(M\) of a Stein manifold are considered, for which it is proved true the conjecture of Tomassini to the effect that a \(2\)-sphere \(\Sigma\) in \(M\) bounds a \(3\)-ball \(B\Subset{M}\), and functions that are holomorphic on a neighborhood of \(\Sigma\) holomorphically extend to a neighborhood of \(\bar{B}\).
scientific article; zbMATH DE number 6087543
scientific article