Holomorphic extension in unbounded subdomains of Stein surfaces

Abstract Let $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> be a subdomain of a Stein surface with smooth strictly pseudoconvex boundary $$M=\partial \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>=</mml:mo> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> . We are mainly interested in the case that $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> is not relatively compact. Building on work by G. Lupacciolu, we describe the envelope of holomorphy of an arbitrary open subset $$M^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> of M in terms of a modified holomorphic hull of the complement $$A=M\backslash M^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>M</mml:mi> <mml:mo>\</mml:mo> </mml:mrow> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> </mml:math> , designed in order to also reflect the geometry of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> at infinity. As a consequence, we clarify the relation between the envelope of $$M^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>M</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:math> and some notions of core sets of $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Ω</mml:mi> </mml:math> , which are an active topic in recent research.