Holomorphic extension in unbounded subdomains of Stein surfaces

Let \(\Omega \) $\mathrm{\Omega }$ be a subdomain of a Stein surface with smooth strictly pseudoconvex boundary \(M=\partial \Omega \) $M=\partial \mathrm{\Omega }$ . We are mainly interested in the case that \(\Omega \) $\mathrm{\Omega }$ is not relatively compact. Building on work by G. Lupacciolu, we describe the envelope of holomorphy of an arbitrary open subset \(M^*\) $M^{\ast }$ of M in terms of a modified holomorphic hull of the complement \(A=M\backslash M^*\) $A=M\backslash M^{\ast }$ , designed in order to also reflect the geometry of \(\Omega \) $\mathrm{\Omega }$ at infinity. As a consequence, we clarify the relation between the envelope of \(M^*\) $M^{\ast }$ and some notions of core sets of \(\Omega \) $\mathrm{\Omega }$ , which are an active topic in recent research.