Envelope of truncated tubes and special domains in higher complex dimensions

TL;DR

This paper introduces special domains in higher complex dimensions, explores their geometry, and generalizes existing results on envelopes of holomorphy and schlichtness from lower to higher dimensions.

Contribution

It defines special domains, proves a theorem on their envelopes in ^n, and extends results on schlichtness and envelopes from ^2 to higher dimensions.

Findings

Every pseudoconvex truncated tube domain is a special domain.

Theorem on the envelope of special domains in ^n for n  2.

Higher-dimensional generalizations of schlichtness results.

Abstract

In this article, we introduce special domains and discuss the geometry of these domains, which includes showing that every pseudoconvex truncated tube domain is a special domain. Next, we prove a theorem for the envelope of special domains in $\C^n ~(n\geq 2)$. Our theorem on special domains is a generalization of a recent result by Jarnicki-Pflug on the envelope of holomorphy of truncated tube domains in $\C^n$. We also establish a result on schlichtness in complex dimension 2, and conclude this article with two higher-dimensional generalizations of the same result by Jarnicki-Pflug mentioned above.