Levi and Trépreau Theorems for Continuous Graphs

Let $\Gamma\subset\mathbb C^{n+1}$ be a continuous graph over a convex domain $D\subset\mathbb C^n\times\mathbb R$ and $a\in\Gamma$ be a point such that none of the components of $(D\times\mathbb R)\setminus\Gamma$ is extendable holomorphically to $a$. Then, $a$ is contained in an $n$-dimensional holomorphic graph lying on and closed in $\Gamma$. In particular, if $\Gamma$ divides two domains of holomorphy, then it is foliated by a family of closed holomorphic hypersurfaces–graphs. These results extend and generalize the well-known theorems of E. Levi, J.-M. Trépreau (proved for $C^2$-smooth $\Gamma$), and N. Shcherbina (proved for $n=1$).