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

2020/07/09 by Noguchi, Junjiro · 1 citation
#2020: 32D10 #32A45 #32Q02 #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2007.04597

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 \mathfrakD:=A₁+iA₂ → Cⁿ with unramified real domains A_j → Rⁿ 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 Cⁿ for which Abe's theorem no longer holds. The present method may clarify the point where the (affine) convexity comes from.

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• Envelope of truncated tubes and special domains in higher complex dimensions

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