A brief proof of Bochner's tube theorem and a generalized tube

TL;DR

This paper provides a new, concise proof of Bochner's Tube Theorem using Oka's Boundary Distance Theorem and extends the theorem to a broader context, including a discussion on convexity and a counterexample to a generalization.

Contribution

It introduces a novel, streamlined proof of Bochner's Tube Theorem and generalizes related convexity results for unramified domains, also providing a counterexample to a proposed extension.

Findings

A new proof of Bochner's Tube Theorem using Boundary Distance Theorem.

Extension of convexity results to unramified domains with pseudoconvexity.

Counterexample showing limits of Abe's generalization.

Abstract

The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be connected by a line segment. We then apply the same idea to show that if an unramified domain $\mathfrak{D}:=A_1+iA_2 \to \mathbf{C}^n$ with unramified real domains $A_j \to \mathbf{R}^n$ is pseudoconvex, then the both $A_j$ are univalent and convex (a generalization of Kajiwara's theorem). From the viewpoint of this result we discuss a generalization by M. Abe with giving an example of a finite tube over $\mathbf{C}^n$ for which Abe's theorem no longer holds. The present method may clarify the point where the (affine) convexity comes from.