Local and global envelopes of holomorphy of domains in $\textbf {C}^ 2$

Abstract:

A criterion is given for a smoothly bounded domain $D \subset {{\mathbf {C}}^2}$ to be locally extendible to a neighborhood of a point ${z_0} \in \partial D$. (This result may also be formulated in terms of extension of CR functions on $\partial D$.) This is related to the envelope of holomorphy of the semitubular domain \[ \Omega (\Phi ) = \{ (z,w) \in {{\mathbf {C}}^2}:\operatorname {Re} w + {r^k}\Phi (\theta ) < 0\} ,\] where $r = |z|$, $\theta = \arg (z)$. Necessary and sufficient conditions are given for the envelope of holomorphy of $\Omega (\Phi )$ to be ${{\mathbf {C}}^2}$. These conditions are equivalent to the existence of a subharmonic minorant for ${r^k}\Phi (\theta )$.