Complex geodesics in convex tube domains

TL;DR

This paper characterizes all complex geodesics in convex tube domains, especially those with unbounded bases, using boundary measures and measure theory, advancing the understanding of holomorphic mappings in these domains.

Contribution

It provides a complete description of complex geodesics in convex tube domains, including cases with unbounded bases, using boundary measure techniques.

Findings

All complex geodesics in convex tube domains are described explicitly.

The description involves boundary measures and real Borel measures on the unit circle.

Results apply to convex tube domains with unbounded bases, including certain Reinhardt domains.

Abstract

We describe all complex geodesics in convex tube domains. In the case when the base of a convex tube domain does not contain any real line, the obtained description involves the notion of boundary measure of a holomorphic map and it is expressed in the language of real Borel measures on the unit circle. Applying our result, we calculate all complex geodesics in convex tube domains with unbounded base covering special class of Reinhardt domains.