Treffer: Levi-flat extensions from a part of the boundary: Levi-flat extensions from a part of the boundary.

Let \(G\) be a bounded domain in \(\mathbb{C}\times\mathbb{R}\) such that \(G\times i\mathbb{R} \subset\mathbb{C}^2\) is strictly pseudoconvex, and let \(U\) be open subset of boundary \(bG\). The authors show that there is an open subset \(\Omega^U\) of \(\overline G\) with \(\Omega^U\cap bG=U\) such that for every \(\varphi\in C(U)\) there exist two functions \(\Phi^\pm\in C(\Omega^U)\) with \(\Phi^\pm| _U=\varphi\) such that the graphs \(\Gamma(\Phi^\pm)\) (over \(\Omega^U\cap G\)) are Levi-flat, and such that \(\Phi^-\leq\Phi\leq\Phi^+\), if \(\Phi\in C(\Omega^U)\), \(\Phi| _U=\varphi\), and its graph \(\Gamma(\Phi)\) is Levi-flat. The authors also show that \(\Omega^U\) is the maximal domain of Levi-flat extensions of the graph of some continuous function on \(U\), when \(G\) is diffeomorphic to a \(3\)-ball and \(U\) satisfies some topological conditions. A sketch of the proof of the main result is given.