Holomorphic extension and schlichtness on tube manifolds

Sammanfattning: We first investigate the holomorphic extension of some classical and accessible classes of domains in $\mathbb{C}^n$ to know to what extent their envelopes are constructive. We give alternative proofs to some of the classical theorems using only first-principle arguments and without involving higher Stein geometry. Next, we address an open problem asked by M. Jarnicki/P. Pflug and construct a counter-example to provide a negative answer to a related open question asked by J. Noguchi, which asks whether the envelopes of holomorphy of truncated tube domains are always schlicht. We also provide a sufficient condition for schlichtness of a tube domain $X+iY$ in $\mathbb{C}^2$, for which $X\subset \mathbb{R}^2$ is a convex domain consisting of finitely many holes with strictly convex $\mathcal{C}^2$-boundary, and $Y\subset \mathbb{R}^2$ is a convex domain.