The Hartogs phenomenon on weakly pseudoconcave hypersurfaces

Abstract

In 2002, Henkin and Michel proved a local Hartogs phenomenon for real analytic CR functions on real analytic weakly pseudoconcave CR manifolds. The aim of the present article is to remove the assumptions on real analyticity in the case of weakly pseudoconcave hypersurfaces \({M\subset\mathbb{C}^n}\) . If M is a graph of class \({\mathcal{C}^2}\) and n ≥ 3, a global theorem is proved for the extension of holomorphic germs along M. If the appearing domains have nicely shaped boundary, a Hartogs theorem even holds for continuous CR functions, where the difference to the case of holomorphic germs relies on the possible presence of lower-dimensional CR orbits. Levi flat hypersurfaces in \({\mathbb{C}^2}\) require a separate treatment. Here an affirmative answer is given to the question of Tomassini, whether 2-spheres bound 3-balls in M.