Complex geodesics in convex tube domains II

Sylwester Zając

doi:10.1007/s10231-015-0537-4

Abstract

Complex geodesics are fundamental constructs for complex analysis and as such constitute one of the most vital research objects within this discipline. In this paper, we formulate a rigorous description, expressed in terms of geometric properties of a domain, of all complex geodesics in a convex tube domain in \({\mathbb {C}}^n\) containing no complex affine lines. Next, we illustrate the obtained result by establishing a set of formulas stipulating a necessary condition for extremal mappings with respect to the Lempert function and the Kobayashi–Royden metric in a large class of bounded, pseudoconvex, complete Reinhardt domains: for all of them in \({\mathbb {C}}^2\) and for those in \({\mathbb {C}}^n\) whose logarithmic image is strictly convex in the geometric sense.

Highlights

A non-empty open set D ⊂ Cn is a tube domain if for some domain ⊂ Rn one has that D = + iRn
The restriction to convex tube domains with no complex lines is made for several reasons, among which the most important is that it results in every holomorphic map with the image in such a domain admitting the boundary measure ([16, Observation 2.5])
In [16], we demonstrated an equivalent condition for a holomorphic map φ : D → D to be a complex geodesic in a convex tube domain D which is taut

Summary

Introduction

Given a convex domain D ⊂ Cn, we call a holomorphic map φ : D → D a complex geodesic for D if there exists a. The restriction to convex tube domains with no complex lines is made for several reasons, among which the most important is that it results in every holomorphic map with the image in such a domain admitting the boundary measure ([16, Observation 2.5]). In [16], we demonstrated an equivalent condition for a holomorphic map φ : D → D to be a complex geodesic in a convex tube domain D which is taut

Preliminaries

Domains in C2