Holomorphic extension in unbounded subdomains of Stein surfaces

Abstract

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